Gray-box optimization is an approach for making some problem-specific information available to the algorithm while still relying on fitness information as the main guide to an optimum. This approach was shown to be beneficial in various combinatorial optimization tasks and neatly captures the continuum between fully black-box algorithms and tailored algorithms. In this work, we discuss different flavors of gray-box algorithms. We show that RLS can find a proper $2$-coloring in a bipartite graph starting from a random $2$-coloring, in an expected time of $\mathcal{O}(n \log n)$. In contrast, when starting from a proper $n$-coloring, the (1+1) EA cannot find such a coloring except when offered additional guiding on plateaus of the search space. Finally, we show the run time for this setting can be much improved by using gray-box operators.

In this paper, we show that generalized Chebyshev polynomials of the second kind, associated with the root system $A_2$, are not arithmetically exceptional. We achieve this by studying the norms of certain cyclotomic elements that admit a parametrization of finite fields.

We study the typical-code (quenched) behavior of the false-alarm (FA) and missed-detection (MD) error exponents of the Neyman-Pearson test associated with soft covering, complementing the average-code (annealed) analysis that has been carried out in a companion paper [1]. We prove that, as the block-length tends to infinity, for almost every randomly selected fixed-composition codebook, the negative normalized logarithms of both error probabilities converge to their respective average-code exponents. In other words, the error exponents are self-averaging. We then extend the scope and study a mismatched likelihood ratio test that assumes the wrong channel model. Here, we derive the mismatched error exponents, show that self-averaging persists under mismatch, and characterize the degradation. In particular, we characterize the coding rate beyond which the two kinds of error exponents cannot be positive at the same time, which in the matched case, is given by the channel input-output mutual information rate.

In millimeter-wave (mmWave) vehicular networks, dense base station (BS) deployments expand the user association (UA) decision space while dynamic blockages cause link quality fluctuations, posing critical challenges for effective mobility management. Traditional Multi-Armed Bandit (MAB) frameworks assume stationary reward distributions and fail to handle the rapid context-reward mapping shifts caused by vehicle mobility and transient blockages. To address this, we propose Blockage-Aware Non-stationary Dynamic Bandit (BAND), a fully distributed, channel state information (CSI)-free mobility management framework for mmWave vehicular networks, formulating UA as a non-stationary contextual bandit problem, enabling online adaptive optimization without requiring central coordination or offline training. BAND employs a cumulative sum-based change detection (CUSUM-CD) to dynamically narrow the active BS set, reducing exploration overhead while tracking reward distribution shifts. Proactive blockage detection suppresses transient signal degradation in the reward estimation process. Simulations demonstrate over 40% regret reduction and up to 33.1% network communication rate improvement compared with hypercube-based contextual bandit baselines, with robustness validated across varying blockage rates and network configurations.

This paper investigates the Markovian dynamics of open two-state quantum systems defined over the real numbers (rebits). Two main objectives are pursued. First, we present a comprehensive classification of Markovian rebit quantum channels, i.e. one-parameter semigroups of completely positive, trace-preserving (CPTP) maps acting on the rebit state space. We show that a full characterisation of their action can be achieved and that describing these channels as solutions of the GKSL equation allows us to explicitly identify the associated Lindblad generators and conditions for complete positivity. Second, we present an original application of this classification to colour perception. Using a recent model in which perceived colours arise from Lüders measurements on the rebit state space, we show how chromatic distortion induced by a non-neutral illuminant can be modelled by a Markovian rebit channel that progressively diminishes colour distinguishability. Other types of channels could be used to study colour vision deficiencies. These phenomena are illustrated by simulations on digital images, highlighting the relevance of rebit Markovian dynamics in modelling colour vision.

Pulsar wind nebulae (PWNe) interacting with the host supernova remnants (SNRs) can enter the reverberation phase in which reverse-shock-driven compression amplifies the magnetic field and rapidly reprocesses particles, sometimes producing "superefficiency", where the radiative output in a given frequency band exceeds the pulsar's instantaneous spin-down power. We investigate the prevalence of this phenomenon in the Galactic population by modeling PWNe with the hybrid TIDE+L framework, which self-consistently follows dynamical evolution, particle spectra, and emission from radio to PeV energies. We track superefficiency across frequency bands and evolutionary stages, analyzing both individual objects and ensemble properties, including compression-resolved samples and population spectral energy distributions. Superefficiency is most common in the far-infrared, but emerges across frequencies and evolutionary phases. It is enhanced in systems where accumulated low-energy electrons radiate in magnetically amplified nebulae. We predict substantially more superefficient sources than a purely thin-shell model would, with differences ranging from factors of a few in FIR and GeV bands to more than an order of magnitude in several optical/UV/X-ray bands.

The paper introduces the model of a pair of branching processes $\{\boldsymbol{Z_n} = \left(Z_n^{(1)}, Z_n^{(2)}\right), \; n \in \mathbb{N}_0\}$ in a joint random environment. If the environment is fixed then the sequences $\{Z_n^{(1)}, \; n \in \mathbb{N}_0\}$ and $\{Z_n^{(2)},\; n \in \mathbb{N}_0\}$ are independent branching processes in a varying environment. This model is a particular case of a more general model of a multitype branching process in a random environment. We establish the asymptotic relation ${\bf{P}}\left(Z_n^{(1)} > 0, Z_n ^{(2)}>0 \right) \sim C n^{-a}$ as $n \to \infty$, where the parameter $a$ depends only on the correlation coefficient $ρ$ of an increment of a two-dimensional associated random walk.

We analytically evaluate the efficiency of continuous dynamical decoupling (CDD) to curb decoherence in generic qubit setups where diverse sources of noise can be present. Previous theoretical approaches to CDD have mainly focused on its potential to cope with longitudinal fluctuations. Here, the basic scenario tackled with CDD is generalized. Apart from dealing with pure dephasing induced by diagonal noise, we consider the impact of transverse fluctuations, usually present in the practical arrangements. In particular, the implications of anisotropic noisy inputs are studied. Additionally, we analyze the role of the fluctuations in the dressing of the qubit by the CDD field of control: since the driving field is usually switched on through linear ramps of its characteristic parameters, the associated dressing of the original states can be described in terms of noisy Landau-Zener transitions. In our approach, based on a sequence of unitary transformations, the noise entering the system is cast into effective stochastic terms whose spectral characteristics are dependent on the driving parameters. This description allows the design of strategies to mitigate the impact of the fluctuations using controlled changes in the effective-noise properties. Significant robustness of CDD against the generalization of the basic scenario can be achieved through an appropriate choice of the parameters of control.

Distributed optimization problems are pervasive in machine learning and optimal control. In this paper, we study smooth strongly convex distributed consensus optimization problems. We present a distributed optimization algorithm for consensus problems based on the Consensus Augmented Lagrangian Alternating Direction Inexact Newton (C-ALADIN) framework. Our algorithm uses an auxiliary variable to decide when to update second-order information, enabling curvature exploitation without sacrificing global convergence. This contrasts with existing C-ALADIN methods, which require constant Hessian approximations and thus lose numerical advantages. Under smooth strong convexity, the algorithm converges globally, and the auxiliary variable converges sublinearly. Numerical experiments on logistic regression show that our algorithm outperforms baseline methods that use either fixed or updated Hessian information.

Conjugated carbon rings are fundamental skeletons of organic functional materials, and their selective formation is of paramount importance in molecular materials engineering. However, steering the formation and 2D crystallization of conjugated carbon rings on the surface with high chemo- and stereoselectivities remains a great challenge. Here, we report a highly chemoselective [1+1+1+1] cycloaddition of isocyanides on the Au(111) surface, which affords the stereospecific tetraaza[4]radialene products and further enables their long-range-ordered 2D crystallization via enantioselective molecular recognition. Using the progressive annealing method, we found that at room temperature, isocyanides undergo a coordination reaction with Au adatoms to form two-fold symmetric isocyanide-Au-isocyanide complexes. In contrast, gradually increasing the annealing temperature induces the transformation of these complexes and subsequent covalent polymerization, leading to the selective generation of tetraaza[4]radialenes with homotactic configurations. The tetraaza[4]radialenes further assemble into 2D homochiral molecular crystals through enantioselective molecular recognition driven by multiple C-H -- Cl hydrogen-bonding interactions. By combining scanning tunneling microscopy/spectroscopy and non-contact atomic force microscopy, we determined the atomic structure and molecular orbitals of tetraaza[4]radialene, confirming that its four-membered ring adopts a planar geometry with a localized lowest unoccupied molecular orbital. Density functional theory calculations suggest that the [1+1+1+1] cycloaddition process involves stepwise formation of C-C bonds and its high selectivity arises from the spatial steric hindrance. Our findings provide new insights into the selective formation of conjugated rings on surfaces and have implications for engineering 2D homochiral molecular crystallization.

This paper is an expository account of cyclotomic spectra. They are spectra (in the sense of homotopy theory) with additional structure that includes an action of the circle group, which we will denote by $\mT$, for torus. Such objects come up in algebraic $K$-theory and its close relatives topological Hochschild homology $\THH$ and topological cyclic homology $\TopC$. They figure prominently in the recent disproof of the {\TC} for chromatic heights greater than 1 by Robert Burklund, Jeremy Hahn, Ishan Levy and Tomer Schlank . Those authors show that for each $n\geq 1$ and each prime $p$, there is a $p$-local ring spectrum $X$ of chromatic height $n$ such that $L_{K (n+1)}\TopC (X)$ and $L_{T (n+1)}\TopC (X)$ (see \cref{def-KT-KK}) are distinct. The present work is part of my attempt to understand theirs

We develop the theory of "automatic actions": (semi)groups acting by $ω$-regular transformations on an $ω$-regular language, showing that it covers a large class of heretofore-unrelated examples. We focus on the subclass of actions by "bounded" $ω$-regular transformations, those for which the Büchi automata encoding the action do not have two connected non-trivial cycles. We show that, for bounded actions of inverse semigroups, the orbit relation is also $ω$-regular. We deduce a number of corollaries, in particular decidability, for such actions, of minimality, topological transitivity, aperiodicity, and order of elements. More generally, every first-order statement over the space of the action, involving the action of specific semigroup elements as well as the relation "being in the same orbit", is decidable. We also apply this result to the study of Julia sets of post-critically finite polynomials, and show that the encoding of Fatou components is also computable; thus every first-order statement involving intersection, disjointness etc. of Fatou components or their full orbit under the polynomial, is decidable.

We present high-resolution X-ray spectroscopic observations of the cool-core galaxy cluster Abell~1795 obtained with XRISM/Resolve. The cluster was observed with two deep pointings: a 225 ks central exposure and a 113 ks northern exposure, extending to a projected radius of 320 kpc from the cluster center. Single-temperature fits reveal a clear radial gradient in the line-of-sight velocity dispersion, decreasing from 114 $\pm$ 11 km/s in the core to 68 $\pm$ 39 km/s at 320 kpc. The bulk velocities in the central regions are very low (22 $\pm$ 12 and 7 $\pm$ 21 km/s), indicating no significant relative motion between the brightest cluster galaxy (BCG) and the intracluster medium (ICM). Given that the central region includes the southward extending cool gas tail, this result disfavors the ``cooling-wake'' scenario and instead supports an AGN-uplift origin. We find that the nonthermal pressure fraction decreases with radius, from $P_{\rm NT}/P_{\rm T}\approx2\%$ in the core to $\sim0.6\%$ at 330 kpc, suggesting that the northern ICM of A1795 is largely quiescent. Two-temperature and split energy-band (2--4 keV and 6--7 keV) fits identify two gas phases within the central $<1.5'$ region, providing strong evidence for multiphase gas in the cluster core. We detect a $\sim14\%$ resonant suppression of the optically thick Fe XXV $w$ line in the center. Additionally, we observe a significant excess in the Fe XXV $y$ line-flux relative to models. Accounting for uncertainties in the atomic data reduces this discrepancy, suggesting that atomic data uncertainties may contribute to the observed residual flux.

Organizational and logical coupling metrics require reliable identification of unique developers. In OSS, commit metadata is limited to names and emails, and the same developer may appear under multiple aliases, which can distort coupling measurements if de-duplication is missing. We aim to build a scalable and accurate pipeline for OSS developer de-duplication and to provide guidance on choosing a model based on precision vs. computational effort. We use Indel similarity as a baseline, then run an LLM-assisted matching process with manual validation to create a large dataset of duplicate identities. Using this dataset, we train and compare classical ML models of different complexity, evaluating precision along with training and inference time and energy. We expect a high-quality dataset and a benchmark of approaches that clarifies which solutions offer the best trade-off between accuracy and cost for large-scale OSS mining.

This paper presents a formulation for strain localization in softening plasticity based on a deformable Cosserat model. The approach enables the direct use of standard elastoplastic constitutive models formulated for a classical Cauchy continuum, without modifying the stress update algorithm or consistent tangent operator. A key feature of the framework is the strict separation of dissipative and energetic mechanisms: all dissipation is confined to the macro-continuum, while the micro-continuum contributes only through linear elastic terms associated with the director field. As a result, the constitutive structure of the elastoplastic model is preserved, and existing models can be employed as black-box components. The internal length scale arises naturally from the micro-continuum and governs the development, interaction and selection of localization patterns, rather than acting as a diffusive parameter. The formulation is easy to implement within standard finite element frameworks, requiring only additional linear contributions to the residual and tangent operators. The performance of the approach is assessed through benchmark problems involving shallow foundations on soil, a demanding test due to complex and unstable localization mechanisms. Both Tresca and Matsuoka-Nakai plasticity models are considered, including cases with highly unstable post-peak responses. Numerical results show convergence of load-displacement responses, dissipated energy and shear-band patterns upon mesh refinement, even in the presence of nonlinear interacting localization processes. These findings demonstrate a robust and physically consistent approach for the analysis of strain localization in softening plasticity.

In this paper, we derive monotone quantities for harmonic functions on asymptotically flat 3-manifolds with simple topology and nonnegative scalar curvature. These monotone quantities are constant on spatial Schwarzschild manifolds outside rotationally symmetric spheres. To derive monotone quantities, our method is different from the ODE analysis in Xia-Yin-Zhou \cite{Xia} and Mazurowski-Yao \cite{Maz}, we follow the strategy developed in Miao \cite{Miao}. As applications, we recover and generalize some geometric inequalities and mass-capacity inequalities in Miao \cite{Miao} and Oronzio \cite{Oronzio}. Furthermore, we obtain the integral identities for the mass-capacity ratio which is parallel to the results in Miao \cite{Miao}.

We prove that the order of vanishing of a natural twist of the $P$-adic Carlitz zeta values at the positive "even" integers is always $1$. We also obtain the same result for $P$-adic Pellarin $L$-series, and for almost all $P$-adic Dirichlet-Goss $L$-series. To do so, we demonstrate a $P$-adic version of a reduced variant of the class formula for Anderson $t$-modules proven by Anglès, Ngo Dac and Tavares-Ribeiro in 2020.

The main goal in regression modelling consists in approximating the conditional mean of a response given a set of features. A regression function is said to be calibrated if the resulting mean estimates match the true conditional means for almost every set of features. Aiming for calibration seems not achievable in practice as one typically deals with finite samples of noisy observations. A weaker notion of calibration is auto-calibration, and it means that the expectation of responses being given the same mean estimate matches this estimate. This notion is important, e.g., in insurance pricing as it ensures no cross-subsidization between different price cohorts. In this paper, we show that boosting trees can be used to test necessary conditions for calibration and auto-calibration, respectively. The practical relevance of our approach is supported by a numerical example, in which the proposed tests prove to be very powerful on a large insurance dataset.

Instantial neighbourhood logic is a modal language for neighbourhood frames in which formulas can express information about the kinds of worlds occurring inside a neighbourhood of a given world. In this paper, we study a positive, negation-free version of instantial neighbourhood logic with two primitive instantial modalities, one of box-type and one of diamond-type. Since classical negation is not available, the two modalities are treated independently. We introduce the language and proof system of positive instantial neighbourhood logic (PINL) and interpret it over persistent two-sided neighbourhood models. We then define a typed persistent neighbourhood semantics, used as an auxiliary canonical semantics to control witness and co-witness conditions. This yields a truth lemma and the corresponding completeness result of PINL. On the algebraic side, we introduce \(2\)-$\mathrm{DLIO}$s, bounded distributive lattices equipped with two families of instantial operations, as the algebraic semantics of PINL. We prove algebraic soundness and completeness via the Lindenbaum \(2\)-$\mathrm{DLIO}$. Finally, we construct the canonical bitopological PINL-space and show that the algebra of its admissible positive opens is isomorphic to the Lindenbaum \(2\)-$\mathrm{DLIO}$. Thus the paper gives a canonical admissible-open representation of positive instantial neighbourhood logic, providing a first step toward a future duality theory.

Grid-forming (GFM) converter-dominated microgrids are often analyzed using reduced-order phasor-domain electromechanical GFM models, but the validity of these models is often taken for granted. Assuming ideal inner-loop tracking (IILT) of terminal-voltage references, these models neglect the inner-loop and filter dynamics at the electromagnetic-transient (EMT) timescale to simplify stability analysis. This paper argues that such neglected dynamics can destabilize the system, invalidating the stability conclusions drawn from the IILT model. To address this cross-timescale stability issue, we formulate the validity of the IILT stability conclusion as a robust-stability certification problem. The EMT-induced model mismatch between the reduced-order converter model and the actual converter model is represented as a structured uncertainty embedded around the IILT feedback loop. This yields a frequency-resolved interaction index and a structured singular-value sufficient certificate for determining when the stability conclusion of the IILT model can be certified with respect to a prescribed EMT uncertainty weight. The uncertainty weight can be obtained from detailed EMT models or terminal reference-response measurements. Case studies confirm that the proposed certificate correctly certifies model validity and identifies the loss of trustworthiness. We also demonstrate that the measurement-based uncertainty weights closely match the model-based ones, which enables deployment without accessing inner-loop models.

Weyl conformal geometry is the natural underlying geometry of gauge theories of the Weyl group (of dilatations and Poincaré symmetry), such as Weyl quadratic gravity and its generalisation, Weyl-Dirac-Born-Infeld action (WDBI). These gauge theories are Weyl-anomaly-free candidates for quantum gravity. We describe Weyl gauge symmetry from a more familiar Riemannian view of Weyl gauge invariant dressed fields by the Wilson line of dilatations. Weyl geometry can then be seen as Riemannian geometry of non-local dressed metric ($g_{μν}^*$), at the cost of non-commutativity in the UV, also induced by the Wilson line. Then Weyl quadratic gravity and WDBI actions of Weyl geometry, which are Weyl gauge invariant in $d$ dimensions, have the same expression in Riemannian geometry defined by $g^*_{μν}$. This is a non-local map between the two geometries and actions in the symmetric phase. Unlike for the metric, the equation of motion of the Weyl gauge field ($ω_μ$) does not commute with the dressing of the metric. When $ω_μ$ becomes massive and decouples (broken phase), commutativity and Einstein-Hilbert action are recovered.

We study a disorder-free variant of the Sachdev-Ye-Kitaev (SYK) model with dissipation within the Gorini-Kossakowski-Sudarshan-Lindblad formalism. By utilizing the integrability of the clean SYK model, we derive an exact solution in a spectrum-resolved form, i.e., the eigenvalues and corresponding projection superoperators of the Liouvillian for arbitrary system size $N$. We determine the scaling of the gap that governs the long-time decay of the two-point correlation functions. Importantly, the gap does not vanish in the dissipationless limit when the thermodynamic limit is taken first, despite the integrability of the model. This phenomenon, known as anomalous relaxation, suggests a possible connection with chaotic dynamics and quantum Ruelle-Pollicott resonances. We also find several spectral features, such as transitions in the Liouvillian spectrum from complex to real eigenvalues with increasing dissipation strength, as well as the convergence of the dissipative form factor to the spectral form factor in the dissipationless limit. These findings indicate that the present model offers a useful platform for exploring nontrivial open dynamics of many-body quantum systems.

Let $a,b$ and $n$ be positive integers and let $S=\{x_1, \cdots, x_n\}$ be a set of $n$ distinct positive integers. For $x\in S$, one defines $G_{S}(x)=\{d\in S: d<x, d|x \ {\rm and} \ (d|y|x, y\in S)\Rightarrow y\in \{d,x\}\}$. We denote by $(S^a)$ (resp. $[S^a]$) the $n\times n$ matrix having the $a$th power of the greatest common divisor (resp. the least common multiple) of $x_i$ and $x_j$ as its $(i,j)$-entry. In 1995, Bourque and Ligh showed that the $a$th power GCD matrix $(S^a)$ divides the $a$th power LCM matrix $[S^a]$ in the ring $M_n(\mathbb Z)$ of $n\times n$ matrices over the integers when $S$ is FC. In 2002, Hong proved that such factorization is no longer true when $S$ is gcd closed. In 2008 (resp. 2026), Hong showed that $[S^a]\mid [S^b]$ if $a\mid b$ and $S$ is a divisor chain (resp. an FC set). In this paper, we show that for arbitrary positive integers $a$ and $b$ with $a|b$, the $b$th power matrices $[S^b]$ is divisible by the $a$th power matrix $[S^a]$ if $S$ is a gcd-closed set (i.e. $\gcd(x_i, x_j)\in S$ for all integers $i$ and $j$ with $1\le i, j\le n$) such that the condition $\mathcal G$ is satisfied (i.e., for any $x\in S$, either $G_S(x)$ contains at most one elements, or $G_S(x)$ contains at least two elements and satisfies that $[y_1,y_2]=x$ and $(y_1,y_2)\in G_S(y_1)\cap G_S(y_2)$) for any $\{y_1,y_2\}\subseteq G_S(x)$. This confirms a conjecture of Hong proposed in [S.F. Hong, Divisibility among power GCD matrices and power LCM matrices, {\it Bull. Aust. Math. Soc.} {\bf 113} (2026), 231-243].

The universal solution of the Korteweg-de Vries equation (KdV) introduced by Gurevich and Pitaevskii in order to describe the onset of dispersive shock waves is known to also obey the self-similar reduction of the next member in the KdV hierarchy. We show that, if this common solution obeys some lower order partial differential equation, its differential order must be one, and we provide its local representation as a converging Laurent series depending on both space and time.

We report progress on the ongoing development of microcalorimeter detector technology for the Hot Universe Baryon Surveyor (HUBS) mission. We show the results from testing and characterizing selected pixels in a 10$\times$10 microcalorimeter array. The microcalorimeter is based on a Mo/Cu transition-edge sensor (TES) coupled to an Au absorber. To better understand the properties of the devices, we have first measured the energy resolution of a selected pixel in a TES array of the same design with a pulsed laser system that produces 3 eV photons, and found that individual photon peaks are easily resolved with the TES, indicating good performance. We have then exposed the microcalorimeter array to radiation from a $^{55}$Fe source, and found that the pixels tested show energy resolutions as good as 3.8$\pm$0.2 eV at 5.9 keV. The energy resolution is found to vary monotonically with the bias point for all the devices, showing little evidence for the presence of the so-called excess noise. This is consistent with the results from modeling the measured noise spectrum. The effects of thermal crosstalk are evident, leading to the degradation of energy resolution.

For real algebraic curves in the plane with singularities we investigate the relation between normals and double normals and the critical points of the squared distance function (up to topological equivalence). For the distance to a given point we show that the topological discriminant consists of the (traditional) evolute, together with some distinguished normal lines at algebraic singular points. We pay special attention to the behavior at points, where the curve is smooth, but only $C^1$ embedded. We discus the differences and similarities with the ED-discriminant in the complex theory. We give counting formula's for normals and double normals, relating them with maxima and minima of the distance function.

We present the first unified study of Earth matter-density stratification and Planck-scale quantum-gravity effects in long-baseline neutrino oscillations, treating both as correlated systematic uncertainties within a full three-flavor framework. Neutrino propagation is computed using matrix exponentiation with spatially resolved Preliminary Reference Earth Model (PREM) density profiles, allowing a quantitative assessment of matter-profile effects beyond the constant-density approximation. We find that the resulting bias in the reconstructed CP-violating phase remains below 0.3° for baselines (L \lesssim 5000) km, but increases to 17.8° at (L=7000) km and 172.2° at (L=12000) km, leading to a potential sign reversal of the inferred CP violation. We further incorporate Planck-scale corrections through the unique gauge-invariant dimension-5 operator of the Standard Model effective field theory. For quasi-degenerate neutrino masses ((m \sim 2) eV), the solar mass-squared splitting receives a correction of order ((1.0 \pm 0.5)\times10^{-5},\mathrm{eV}^2), while the atmospheric splitting remains essentially unchanged. Most importantly, we identify a previously unexplored degeneracy regime in which Planck-scale perturbations can partially compensate matter-induced biases. Around (L\approx7000) km, specific Majorana phases reduce the combined bias by about 30%, demonstrating that the two effects cannot be treated independently. These results highlight the necessity of propagating both Earth-density and quantum-gravity uncertainties as correlated systematics in precision measurements of leptonic CP violation.

In the category of topological Abelian groups, we consider the usual notion of an extension $E=(B \to X \to A)$ of $B$ by $A$, together with the notion of a weakly split extension, i.e., an extension for which the projection $X \to A$ admits a continuous section $A \to X$. Given a weakly split extension $E$, the topological Abelian group $X$ is homeomorphic to $B \times A$, although in general it is not algebraically isomorphic to $B \times A$. For two topological Abelian groups $A$ and $B$, we study the Abelian group $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ of weakly split extensions of $B$ by $A$, modulo extension isomorphisms. We show that $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ can be described as the group of all continuous sum structures defined on the product space $B \times A$ (up to topological isomorphism), with $B$ as a topological subgroup and $A$ as a topological quotient. We also provide an alternative description of $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ as a quotient $Z_c(A,B)/B_c(A,B)$, where $Z_c(A,B)$ consists of cocycles given by continuous maps $A \times A \to B$, and $B_c(A,B)$ denotes the corresponding coboundaries. Furthermore, we compare $E^{\mathrm{ws}}_{\mathrm{TA}}(A,B)$ with the group of standard extensions $E_A(A,B)$, where $A$ and $B$ denote the underlying Abelian groups, and relate these constructions by means of a six-term exact sequence. Although the Bohr topology of discrete Abelian groups has been investigated by many workers, there still remain many parts that are not well understood. Here, as an application of the methods developed in the paper, new examples of nontrivial $ws$-extensions for discrete Abelian groups equipped with the Bohr topology are provided and some related open questions are also proposed.

Stimulated by Aldroubi and his collaborator's recent work on dynamical sampling, we consider a homogeneous discrete dynamical system of the form $f_n = A f_{n-1}= A^{n}f, \quad f_0 = f$, where A is a bounded linear operator on a separable Hilbert space H, which is known as the evolution operator, and $f_0 \in \mathcal{H}$ is the unknown initial state. The associated dynamical samples are given by the collection $\{\langle A^n f, g\rangle: g \in \mathcal{G, 0 \leq n < n < L(g)}\}$, where $\mathcal{G} \subset H$, is a finite or countable sampling set and $L$ is a function $L: G \rightarrow \mathbb{N} \bigcup \{\infty\}$. We analyze the stability of perturbed dynamical sampling systems in the sense of Nagy and Pollard-Hilding. More precisely, we establish sufficient conditions for the stable recovery of an initial state from perturbed dynamical samples obtained by changing the sampling vector, the evolution operator, ro simultaneously both, within the framework of Nagy-type and Pollard-Hilding-type perturbation of frames.

Objective: To determine whether free-breathing golden-angle radial sparse parallel (GRASP) magnetic resonance imaging (MRI) can represent respiratory-induced organ motion in patients with liver malignancies undergoing stereotactic body radiation therapy (SBRT). Methods: A retrospective analysis of 54 patients undergoing liver SBRT was conducted. Four-dimensional computed tomography (4D-CT), the gold standard for motion assessment, was used to characterize liver tumor motion. Image fusion was performed between free-breathing GRASP MRI and each respiratory phase of 4D-CT using an in-house registration program, with fusion quality quantified by maximum cross-correlation coefficient (MCC). Validation involved two blinded radiation oncologists: one repeated image fusion using the Eclipse-built-in module, while the other evaluated clinical relevance on a five-point scale. Results: The 50% respiratory phase of 4D-CT achieved the highest fusion quality with GRASP MRI, showing no significant differences compared to the 30% (P = 0.106), 40% (P = 0.632), and 60% (P = 0.792) phases. In contrast, fusion quality declined significantly beyond the mid-respiratory window (30%-60%), with poor fusion at the 0%, 10%, 20%, 80%, and 90% phases (P < 0.001). Validation by radiation oncologists corroborated these findings, with the 50% phase achieving the highest score. Subjective scores remained above 4 for phases 30%-70%, while scores for the remaining phases fell below 4. Conclusion: Free-breathing GRASP MRI cannot independently represent organ motion across all respiratory phases; it accurately characterizes motion only within the mid-respiratory phases (30%-60%), with optimal performance at the 50% phase. When used as a delineation standard in liver SBRT, GRASP MRI should be combined with 4D-CT or dynamic imaging modalities to ensure comprehensive motion assessment and accurate target volume definition.