Debris disks are created and sculpted by planetary bodies in the orbital space they share. The properties of these disks, including mass, orbital extent, and morphology, can be indicators of their planetary shepherds. Recently, T. Pearce and collaborators placed limits on the masses and orbits of hypothetical planets around 178 stars with resolved debris disks. We consider 176 of these stars, all the objects that have astrometric data in the Gaia Data Release 3 archive, to assess planet detection from astrometry. Our analysis begins with a set of stellar hosts of known exoplanets, selected to roughly match the parallax, apparent magnitude, and color of the 176 debris disk systems. We confirm that Gaia's ruwe parameter, a measure of the quality of astrometric fitting to a linear drift model, is sensitive to the presence of massive companions, even planetary ones. Guided by ruwe and a metric derived from a machine-learning algorithm trained on Gaia parameters from the exoplanetary host data set, we identify promising stars with debris disks that may host as-yet-undiscovered planets. These stars will be compelling subjects for time-series analyses with Gaia Data Release 4.

As future tasks in networked systems are increasingly relying on collaborative execution among distributed devices, trust has become an essential tool for securing both reliable collaborators and task-specific resources. However, the diverse requirements of different tasks, the limited information of task owners on others, and the complex relationships among networked devices pose significant challenges to achieving timely and accurate trust evaluation of potential collaborators for meeting task-specific needs. To address these challenges, this paper proposes Trust-as-a-Service (TaaS), a novel paradigm that encapsulates complex trust mechanisms into a unified, system-wide service. This paradigm enables efficient utilization of distributed trust-related data, need-driven trust evaluation service provision, and task-specific collaborator organization. To realize TaaS, we develop an agentic AI-based framework as the enabling platform by leveraging the Model Context Protocol (MCP). The central server-side agent autonomously performs trust-related operations in accordance with specific task requirements, delivering the trust assessment service to all task owners through a unified interface. Meanwhile, all device-side agents expose their capabilities and resources via MCP servers, allowing devices to be dynamically discovered, evaluated, engaged, and released, thereby forming task-specific collaborative units. Experimental results demonstrate that the proposed TaaS achieves 100\% collaborator selection accuracy, along with high reliability and resource-efficient task completion.

This paper presents TSO-DSO coordinated reactive power dispatch, with a focus on real-time implementation. A sensitivity-aware, mixed-integer linear programming (MILP) formulation is developed to model the IEEE 1547-compliant droop-based control modes Volt VAR (VV), Volt Watt (VW), and Watt VAR (WV) of smart inverters. The algorithm employs a hierarchical optimization strategy using Special Ordered Sets (SOS1) to enhance computational efficiency and supports limited measurement scenarios through Recursive Least Squares (RLS) estimation. The proposed method is tested on the IEEE 13-bus and 123-bus distribution networks, which are connected to a 9-bus transmission system. Results demonstrate the feasibility and effectiveness of the real-time dispatch framework in improving voltage regulation and minimizing power curtailment.

Interactions and time shape many aspects of life. Everyday activities -- like conversations, emails, money transfers, citations, and even acts of violence -- are relational events: interactions between a sender and a receiver at a specific moment. At the intersection of event-history analysis and network modelling, relational event models (REMs) offer a powerful framework for studying when and why these events occur. Recent advances have made it possible to express REMs as generalized additive models, allowing researchers to capture complex, non-linear patterns over time. While an essay and a comprehensive review exist, a hands-on tutorial paper on REMs is still missing. This work fills that gap. It provides a practical introduction to REMs, incorporating the latest developments in the field. It demonstrates how to simulate synthetic relational-event data and walks through several empirical applications, comparing different modelling and inference strategies. By bringing together theory, simulation, and application, this tutorial lowers the barrier to entry and makes REMs a more accessible and practical tool.

We classify $n\times n$-matrix-valued continuous commutativity and spectrum preservers defined on spaces of (a) normal, (b) semisimple and (c) arbitrary $n\times n$ matrices with spectra contained in sufficiently connected subsets $\mathcal{X}\subseteq \mathbb{C}$, generalizing a number of results due to Šemrl, Gogić, Tomašević and the author among others. In case (a) these are always conjugations or transpose conjugations, while in cases (b) and (c) qualitatively distinct possibilities arise depending on the local regularity of the complex-conjugation map close to coincident-eigenvalue loci of $\mathcal{X}^n$.

Topological defects (TDs) are crucial for understanding important physical properties of crystalline materials including mechanical failure, ion transport, and two-dimensional melting. This concept has not translated to disordered materials like glasses because these solids have no obvious reference structure that can be used to define TDs. As a result, key properties related to those listed above have typically been modeled using purely phenomenological approaches. Recent studies have demonstrated that certain observables commonly associated with TDs can also be identified in disordered solids indicating that topological concepts may be as crucial in amorphous solids as in crystals. This hints that TDs may offer a first-principles framework for understanding their mechanical response and complex spatiotemporal dynamics. In this Perspective, we review recent theoretical, numerical, and experimental studies that have exploited topological concepts to rationalize mechanical properties of amorphous solids. We also highlight pressing open questions and some promising directions for future research in the field.

Based on experimental results, this work proposes the influence of the Oxygen order present in the Ortho-II superstructure of YBa$_2$Cu$_3$O$_{7-δ}$ (YBCO), on the final ordering of Oxygens in its Orthorhombic phase for $δ$ $\approx$ 0. Isothermal oxygenation (oxyg) of YBCO powder material is performed, starting from non-oxygenated material ($δ$ $=$ 1) and evolving until saturation in an oxygen atmosphere. The oxyg process is carried out within a temperature range from 300 $^o$C to 800 $^o$C (300 $^o$C $<$ T$_O$ $<$ 800 $^o$C). During the oxyg process, and using a thermogravimetric balance, the evolution of mass (m) and the differential thermal analysis (DTA) of the material are monitored with respect to an inert reference material subjected to the same conditions as the YBCO powder. These results allow observation of the Tetragonal-Orthorhombic (T-O) transition occurring in the YBCO material. From these results, oxygenated YBCO material is obtained by working at different temperatures and under two different conditions: through a direct T-O transition into the Ortho-I superstructure, and by passing through the Ortho-II superstructure along the transition. The material obtained under these two conditions is studied by X-Ray diffraction, revealing differences in the resulting diffractograms. Furthermore, we propose that, for low values of T$_O$ (T$_O$ $<$ 400 $^o$C), the T-O transition proceeds through the region of the phase diagram where the Ortho-II superstructure is present, leading to progressive ordering of the Oxygen atoms within the material. This ordering leaves a fingerprint in the final configuration reached by the YBCO material, even beyond the region where the Ortho-II superstructure is stable. Finally, we suggest that this mechanism is responsible for the differences observed between the diffractograms obtained under both conditions.

Generalized finite automata (GFAs), probabilistic finite automata (PFAs), and one-way general quantum finite automata (1gQFA) recognize the same strict-cutpoint languages, but the state complexity of exact probabilistic simulation has remained unclear. This paper determines that worst-case cost exactly: every \(n\)-state 1gQFA admits exact strict-cutpoint simulation by a one-way PFA with \(O(n^2)\) states, via the standard \(n^2\)-dimensional mixed-state linearization together with an explicit alphabet-preserving construction that converts each \(k\)-state GFA into a one-way PFA with at most \(2k+6\) states; conversely, for every \(n\ge 2\), there exists an \(n\)-state 1gQFA for which every equivalent one-way PFA requires at least \(n^2-1\) states, obtained from a prepare--test construction and a Vapnik--Chervonenkis dimension argument. Hence the worst-case probabilistic state cost of exact strict-cutpoint simulation is \(Θ(n^2)\).

Given a connected reductive algebraic group $G$, we consider the class of spherical subgroups $H \subset G$ such that $H$ is regularly embedded in a parabolic subgroup $P \subset G$ and $H,P$ have a common Levi subgroup $L$. In a previous paper, the author developed a fast algorithm that reduces the computation of the set of spherical roots for such subgroups $H$ to the case where the quotient of Lie algebras $\operatorname{Lie} P / \operatorname{Lie} H$ is a strictly indecomposable spherical $L$-module. In this paper, we complete the classification of all such cases and compute the spherical roots for each of them, which enables one to use the above fast algorithm directly for computing the spherical roots for arbitrary spherical subgroups in the class under consideration.

The space-time correlation of a passive scalar advected by a Gaussian colored-noise velocity with wavenumber-dependent correlation times and power-law spatial spectra is investigated in the present paper. Within the inertial-convective subrange, we derive an analytical solution for the space-time correlation. This solution validates the elliptic approximation (EA) model [He and Zhang, Phys. Rev. E 73, 055303(R) (2006)], demonstrating that the iso-correlation contours are self-similar in the co-moving space-time frame $(r-Uτ, Vτ)$, with a universal spatial-to-temporal intercept ratio of 1.55. Unlike the classic Kraichnan white-noise model, our formulation simultaneously recovers the Obukhov--Corrsin scaling for spatial correlations (when the velocity obeys Kolmogorov scaling) and reproduces the random-sweeping mechanism, yielding Gaussian (rather than exponential) temporal decorrelation of scalar Fourier modes. Our results clarify the underlying decorrelation mechanism of passive scalars: mean-flow advection and large-scale sweeping dominate temporal decorrelation, and small-scale distortion dominates spatial decorrelation.

Existing short term voltage stability (STVS) methods typically address either voltage oscillations or delayed voltage recovery; however, the coexistence of both phenomena has not been adequately covered in the literature. Moreover, existing real-time STVS assessment methods often provide only binary stability classifications. This paper proposes novel indices that enable early detection and quantify the degree of stability. The proposed method decomposes post-fault voltage trajectories using Empirical Mode Decomposition (EMD) into residual and oscillatory components. It then employs Lyapunov Exponents (LEs) to characterize the dynamic behavior of each component and evaluates the stability degree using Kullback Leibler (KL) divergence by comparing the LEs of each component with those of a predefined critical signal. The proposed indices assess oscillatory stability significantly faster than the traditional LE method applied directly to the original signal. Specifically, they detect stability within 0.6 seconds after a fault, compared to approximately 10 seconds for the conventional LE approach. In addition, the delayed-recovery index can identify generator trips caused by over-excitation limits within 3 seconds, well before the actual trip occurs at approximately 20 seconds, thereby providing operators and controllers sufficient time to take preventive actions. Furthermore, thresholds are derived to distinguish between stable and unstable cases, offering a graded measure of the stability margin. Simulation studies on the Nordic test system under varying load conditions demonstrate the effectiveness of the proposed indices.

In a recent paper [M. Gül et al., Phys. Rev. E, 110 (6), 064115] we showed that test particle sum rules, which address the excess chemical potential and isothermal compressibility, could be used to develop new and accurate classical density functionals for hard-sphere (HS) fluids. Here we extend our approach to the construction of HS functionals building upon the state of art White-Bear (WB) and White-Bear mark II functionals. Employing the same test-particle sum rules we determine the two free parameters in the Lutsko [James F. Lutsko, Phys. Rev. E, 102, 062137] formulation of fundamental measure theory (FMT) by minimizing the relative errors between different routes to the two thermodynamic quantities. The resulting optimized Lutsko WB functionals, especially Lutsko WB mark II, are generally more accurate and consistent than those obtained in earlier treatments.

In this paper, we study the internal structure of light pseudoscalar mesons using spin improved power-law wave functions. We choose the pion and the kaon for our work. We use the standard quark-quark correlation functions to calculate the distribution amplitudes (DAs), parton distribution functions (PDFs), transverse momentum dependent parton distribution functions (TMDs), and generalized parton distribution functions (GPDs) at zero skewness and form factors. We present all the above distribution functions through the overlap of light-front wave functions (LFWFs). We use leading-order Efremov-Radyushkin-Brodsky-Lepage (ERBL) equations for DAs and next-to-leading-order (NLO) Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations for PDFs to evolve them to higher scales. We find that only 41\% of the longitudinal momentum fraction is carried by the quark and antiquark of both pion and kaon at 16~GeV$^2$. The vector form factors for both the pion and the kaon are found to be in good agreement with experimental data. Similarly, the electromagnetic charge radii are found to be 0.668~fm and 0.704~fm for the pion and kaon, respectively.

We establish the Hasse principle for $100\%$ of conic bundles over $\mathbb{P}^1_{\mathbb{Q}}$.

The $\vartheta$-MGGA class of density functionals is formally reformulated as Hessian-level meta-generalized gradient approximations (HL-MGGAs). In contrast to standard meta-GGAs that rely on the orbital-dependent kinetic-energy density or the density Laplacian, HL-MGGAs utilize the full density Hessian. We introduce a simplified, non-empirical functional, $\vartheta$-PBE, and present a roadmap for its self-consistent implementation within the projector augmented-wave (PAW) method. By utilizing the complete set of spatial second-order density derivatives, the functional's underlying descriptor successfully distinguishes between distinct one-electron density limits, such as single-center atomic densities and two-center bonds, that standard iso-orbital indicators often conflate. Benchmarks across molecular and solid-state datasets reveal that while $\vartheta$-PBE delivers accurate chemisorption energies and molecular properties, challenges remain in predicting bulk lattice constants. Ultimately, this work demonstrates the physical utility and feasibility of designing orbital-independent, Hessian-based exchange-correlation functionals.

The reconfigurable intelligent surface (RIS) has attracted considerable attention of both academia and industry in recent years, given its capacity to dynamically manipulate the reflection of incident electromagnetic waves. Although the research developed for the RIS may have reached its maturity, there are still contentious aspects and limitations regarding its potential benefits for the next generation of wireless communications. In order to improve upon the the RIS technology, the beyond diagonal reconfigurable intelligent surface (BD-RIS) was recently proposed as an promising alternative. The BD-RIS boasts a more sophisticated circuit topology that is capable of providing more combinations of different adjustments or configurations for signal reflection. However, to aptly reap the benefits of the BD-RIS, the added degrees-of-freedom of its configuration must be leveraged accordingly. Therefore, in this work we propose a depth-first tree search algorithm for configuring the BD-RIS in multi-user multiple-input single-output (MU-MISO) communication systems. Taking advantage of the tree search exploration, the proposed algorithm achieves a remarkable trade-off between channel strength maximization performance and computational complexity scalability.

The relic abundance of vector dark matter originating from an inherited axion-$SU(2)$ condensate is typically determined by implementing an adiabatic matching procedure across the symmetry-breaking transition. We demonstrate that this outcome does not arise in the generic case. The post-inflationary crossover can instead be formulated as a dynamical quantum quench problem, in which the residual coherent component of the field is characterized by a survival factor that induces an $\mathcal{O}(1)$ renormalization of the standard abundance relation. Expressed in conformal time, the spatially homogeneous condensate dynamics reduce to those of a canonical oscillator with quartic and quadratic self-interactions. This representation enables an analytic determination of the matching conditions across the symmetry-breaking transition, the derivation of the corresponding quench work and excess energy relations, and a quantitative validation of the coherent sector description via numerical simulations in both Minkowski and Friedmann--Robertson--Walker backgrounds. We also formulate the homogeneous fluctuation theory via the diagonal-$SO(3)$ $1 \oplus 3 \oplus 5$ decomposition and isolate a soft traceless-symmetric quintet with a $k=0$ vacuum obstruction, a regulated ultraviolet adiabatic bound, and a positive quartic stabilization term. Collectively, these results refine the theoretical description of inherited non-Abelian dark matter production and establish the necessary infrared framework for subsequent investigations of finite-$k$ gauge--Higgs transfer dynamics.

This paper investigates the role of complex and negative roots in mixed causal-noncausal autoregressive (MAR) models. Using partial fraction decompositions, we show that seasonal roots can always be isolated in the moving average representation of purely causal and noncausal AR models. We find that this result extends to the MAR model, which means that no new joint seasonal effects can be generated despite the multiplicative structure of the causal and noncausal polynomials. This results has important consequences for the MAR model selection procedure and these are extensively studied in a Monte Carlo simulation study. An empirical application on COVID-19 and soybean data illustrates the main findings of the paper.

We construct a causal and covariantly stable kinetic model whose spectrum at real wavenumbers $k$ reproduces any rest-frame stable dissipative dispersion relation $ω(k)$ via suitable initialization of the microscopic degrees of freedom. Macroscopic observables can therefore obey arbitrary linear evolution equations (including forms that would be acausal if taken as fundamental), while the underlying dynamics remains causal, and all apparent propagation is encoded in the initial data. This provides an explicit counterexample to the idea that microscopic causality alone constrains the analytic form of dispersion relations at real $k$. In particular, bounds on transport coefficients based solely on the analytic structure of $ω(k)$, such as the hydrohedron bounds, require additional assumptions about the region in the complex $k$-plane where $ω(k)$ corresponds to physical modes.

Bicycle node networks are regional bicycle networks equipped with a wayfinding system of numbered nodes to ease recreational cycling. They spur sustainable bicycle tourism, economic spending, and local culture. Due to their country-wide scale, implementing bicycle node networks is a considerable effort and investment. Despite this investment, planning is a manual ad-hoc process that follows general design principles, but without clear performance metrics that account for the human cycling experience. Here we analyze a 28,215 km long bicycle node network spanning Denmark, developing and studying such metrics. First, a spatial analysis of geometric and topological properties reveals high heterogeneity and local clusters of node density, face loop lengths, gradients, and feature-rich areas. Next, taking the perspective of a recreational cyclist starting at any node on the network, we create a loop census that lists all loops in the network up to day-trip length. The loop census identifies the feasible points on the network from which to take a day trip and quantifies the number of round trip choices, unveiling different levels of choice depending on the considered demographic group. While long-range cyclists can access most of the country with often overabundant choices, cyclists with stronger length and gradient limitations like families with small children can not - which could be overcome by e-bikes. Our open-source analysis methods provide data-driven decision support for bicycle node network planning with the potential to boost the development of rural cycling and cycling tourism.

Dust torus plays the key role in determining active galactic nuclei (AGN) observational appearance. Here, the scenario of accretion regulated central dust torus is tested for the first time in the individual changing-look AGN (CLAGN) SDSS J1011+5442. Through the dependence of broad H$α$ luminosity on continuum luminosity, the scenario of moving dust clouds can be ruled out in SDSS J1011+5442. Meanwhile, virial BH mass in the bright state is consistent with the M-sigma relation determined mass, indicating the virialization assumptions efficient in central BLRs. However, the virial BH mass determined in the dim state is 60 times smaller than the M-sigma relation determined value. The contrary properties of broad H$α$ in different states can be naturally explained by the scenario of accretion regulated dust torus. Below a critical Eddington ratio, opening angle of dust torus declines with increasing accretion rate, leading to only outer part of central BLRs for broad H$α$ with smaller line widths detected in the dim state but all the BLRs detected in the bright state. The results in this manuscript not only indicate properties of central dust torus having apparent effects on variability properties of CLAGN, but also indicate that studying CLAGN could provide further clues to check dynamical evolving models for dust torus in AGN.

We study set-valued decision rules in which performance is defined by the inclusion of the top-$p$ hypotheses, rather than only the single best or true hypothesis. This criterion is motivated by sensor selection for target tracking, where inexpensive measurements are used to identify a list of sensor nodes that are likely to be closest to a target. We analyze the performance of top-$p$ versus top-$1$ selection under sequential hypothesis testing, propose a geometry-aware sensor selection algorithm, and validate the approach using real testbed data.

Time series graphical models have recently received considerable attention for characterizing (conditional) dependence structures in multivariate time series. In many applications, the multivariate series exhibit variable-partitioned blockwise dependence, with distinct patterns within and across blocks. In this paper, we introduce a new class of time series Gaussian chain graph models that represent contemporaneous and lagged causal relations via directed edges across blocks, while capturing within-block conditional dependencies through undirected edges. In the frequency domain, this formulation induces a cross-frequency shared group sparse plus group low-rank decomposition of the inverse spectral density matrices, which we exploit to establish identifiability of the time series chain graph structure. Building on this, we then propose a three-stage learning procedure for estimating the undirected and directed edge sets, which involves optimizing a regularized Whittle likelihood with a group lasso penalty to encourage group sparsity and a novel tensor-unfolding nuclear norm penalty to enforce group low-rank structure. We investigate the asymptotic properties of the proposed method, ensuring its consistency for exact recovery of the chain graph structure. The superior empirical performance of the proposed method is demonstrated through both extensive simulation studies and an application to U.S. macroeconomic data that highlights key monetary policy transmission mechanisms.

We take causality and uniqueness of events observation as our driving forces. They are built in in the way we define distinct observers, which then require a finite time to communicate between each other. This unavoidably leads to the existence of maximal transfer-information velocity between arbitrary (not necessarily inertial) reference frames. Inertial reference frames are defined by fixing the geometrical properties of (spatial) distance without any reference to relativity, electromagnetism, or laws of physics in general. For these inertial reference frames, the causality condition fixes the causal group to be the orthochronous inhomogeneous Lorentz group times dilatations. The mathematics we will use are quite basic.

The concept of freely falling frames suggests that gravity exhibits a local Lorentz gauge symmetry and requires a background Minkowski reference frame. The gauge vector fields of a Yang-Mills-type theory can be constructed from the transformations to these local freely falling frames, thereby leading to a composite theory of gravity. We propose coordinate conditions under which an exact black-hole solution can be obtained. Our analysis of planar gravitational waves reveals that, despite the large symmetry group, composite gravity possesses only four physical degrees of freedom. We elaborate a Hamiltonian formulation of composite gravity, derive the full set of constraints for the nonlinear theory, and outline the pathway toward its quantization.

The first measurement of inclusive J/psi polarization at mid-rapidity (|y^{J/psi}| < 0.8) in 200 GeV Ru+Ru and Zr+Zr collisions at sqrt(s_NN) = 200 GeV with the STAR experiment at RHIC is presented. J/psi mesons are reconstructed through their di-electron (e+e-) decay channel. The polarization parameters (lambda_theta, lambda_phi) are measured as a function of the J/psi transverse momentum (p_T) and collision centrality in both the helicity and the Collins-Soper frames. These polarization parameters are found to be consistent with zero across the measured J/psi p_T range of 0.2 < p_T < 10 GeV/c and across collision centralities within 0-80 percent in both frames. These results are consistent with corresponding measurements p+p collisions at the same collision energy and with transport-model calculations.

Time-varying distortions in communication systems can significantly degrade the performance of soft-decision forward error correction. This paper presents a burst-aware (BA) low-density parity-check (LDPC) decoding scheme for channels affected by bursty phase noise. By applying differential coding to a Wiener process with time-varying innovation variance, bursty differential phase noise is obtained. Simulation results demonstrate that, compared to conventional decoding, the BA scheme achieves gains in the signal-to-noise ratio of up to $0.7$~dB at a bit error rate (BER) of $4\cdot10^{-3}$ and more than $1$~dB at a packet error rate (PER) of $1\cdot10^{-2}$. Furthermore, by iterating between channel estimation and \ac{ldpc} decoding, forming the proposed iterative burst-aware (IBA) decoding scheme, the gains increase to $1.4$~dB and more than $3$~dB, respectively. More importantly, the IBA scheme significantly improves robustness to bursty phase noise. Compared with the conventional scheme, the IBA scheme can reduce both \ac{ber} and \ac{per} by up to two orders of magnitude under severe bursty phase noise.

We study an overdetermined elliptic free boundary problem for exterior domains in $\mathbb{R}^N$, $N \ge 2$, introduced by F. Morabito [Comm. PDE 46 (2021), 1137-1161]. The overdetermining condition prescribes the Neumann data as a multiple of the boundary mean curvature, with parameter $Γ$, together with a spherical compatibility condition. For $N \ge 3$, we prove rigidity of the spherical solution among star-shaped domains when $Γ\ge N-2$; in the borderline case $Γ= N-2$, the star-shapedness assumption can be removed, and rigidity holds among all bounded domains. The proof combines the Pohozaev identity, geometric identities, and the sharp boundary inequality of Agostiniani and Mazzieri for capacitary potentials. We also obtain rigidity among bounded domains for $Γ\le 0$ via Serrin's moving plane method. In dimension two, the unit disc is the only admissible domain for every $Γ$.

We prove that $\mathrm{SL}_3(\mathbb{Z})$ contains a non-split sharply 2-transitive subgroup, answering a question of Glasner and Gulko. We also prove that $\mathrm{SL}_4(\mathbb{Z})$ contains a non-split sharply 3-transitive subgroup, but that $\mathrm{SL}_3(\mathbb{Z})$ does not contain an infinite sharply 3-transitive subgroup.

A graph $G$ is $k$-vertex-critical if $χ(G)=k$ but $χ(G-v)<k$ for all $v\in V(G)$. In this paper we make progress on the open problem of the finiteness of $k$-vertex-critical $(P_4+\ell P_1)$-free graphs by showing that there are only finitely many $k$-vertex-critical graphs in the following subfamilies of $(P_4+\ell P_1)$-free graphs for all $k\ge 1$ and $\ell\ge 0$: $\bullet$ $(P_4+\ell P_1,\text{chair})$-free graphs, $\bullet$ $(P_4+\ell P_1,P_5,\text{bull})$-free graphs, and $\bullet$ $(P_4+\ell P_1,P_5,\text{cricket})$-free graphs. In fact, all but the first of these are special cases of our general result that there are only finitely many $k$-vertex-critical $(P_4+\ell P_1,B_{4}(m),B_{3}(m)^{+})$-free graphs for all $k\ge 1$ and $\ell,m\ge 0$. Here $B_{n}(m)$ is the graph obtained from a path of order $n$ by identifying one of its leaves with the centre vertex of $K_{1,m}$ and $B_{n}(m)^{+}$ is the graph obtained by identifying an edge of $K_3$ with the edge of $B_{n}(m)$ with endpoints of degrees $2$ and $m$, respectively. Our results imply the existence of simple polynomial-time certifying algorithms to decide the $k$-colourability of all graphs in these subfamilies for every fixed $k$. We also show that $χ(G)\le \ell+2$ for all $(P_4+\ell P_1,K_3)$-free graphs and all $\ell\ge 0$, improving the previously known upper bound of $2\ell+2$ that followed from Randerath and Schiermeyer's 2004 result on $(P_t,K_3)$-free graphs. More generally, we provide a $χ$-bound in $O(\ell^{ω-1})$ for $(P_4+\ell P_1)$-free graphs which improves the bound of $(2\ell+2)^{ω-1}$ which followed from Gravier, Hoàng and Maffray in 2003 for $P_{t}$-free graphs.