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The Aldous-Hoover Theorem concerns an infinite matrix of random variables whose distribution is invariant under finite permutations of rows and columns. It states that, up to equality in distribution, each random variable in the matrix can be expressed as a function only depending on four key variables: one common to the entire matrix, one that encodes information about its row, one that encodes information about its column, and a fourth one specific to the matrix entry. We state and prove the theorem within a category-theoretic approach to probability, namely the theory of Markov categories. This makes the proof more transparent and intuitive when compared to measure-theoretic ones. A key role is played by a newly identified categorical property, the Cauchy--Schwarz axiom, which also facilitates a new synthetic de Finetti Theorem. We further provide a variant of our proof using the ordered Markov property and the d-separation criterion, both generalized from Bayesian networks to Markov categories. We expect that this approach will facilitate a systematic development of more complex results in the future, such as categorical approaches to hierarchical exchangeability., Comment: 39 pages

The LIGO, Virgo and KAGRA gravitational wave observatories are currently undertaking their O4 observing run offering the opportunity to discover new electromagnetic counterparts to gravitational wave events. We examine the capability of the Neil Gehrels Swift Observatory (Swift) to respond to these triggers, primarily binary neutron star mergers, with both the UV/Optical Telescope (UVOT) and the X-ray Telescope (XRT). We simulate Swift's response to a trigger under different strategies using model skymaps, convolving these with the 2MPZ catalogue to produce an ordered list of observing fields, deriving the time taken for Swift to reach the correct field and simulating the instrumental responses to modelled kilonovae and short gamma-ray burst afterglows. We find that UVOT using the $u$ filter with an exposure time of order 120 s is optimal for most follow-up observations and that we are likely to detect counterparts in $\sim6$% of all binary neutron star triggers detectable by LVK in O4. We find that the gravitational wave 90% error area and measured distance to the trigger allow us to select optimal triggers to follow-up. Focussing on sources less than 300 Mpc away or 500 Mpc if the error area is less than a few hundred square degrees, distances greater than previously assumed, offer the best opportunity for discovery by Swift with $\sim5 - 30$% of triggers having detection probabilities $\geq 0.5$. At even greater distances, we can further optimise our follow-up by adopting a longer 250 s or 500 s exposure time., Comment: 16 pages, 13 figures. Final version accepted by MNRAS

We present the results of our multi-wavelength (X-ray to radio) follow-up campaign of the Einstein Probe transient EP240408a. The initial 10 s trigger displayed bright soft X-ray (0.5-4 keV) radiation with peak luminosity $L_\textrm{X} \gtrsim 10^{49}$ ($10^{50}$) erg s$^{-1}$ for an assumed redshift z>0.5 (2.0). The Neil Gehrels Swift Observatory and Neutron star Interior Composition ExploreR discovered a fading X-ray counterpart lasting for $\sim$5 d (observer frame), which showed a long-lived (~4 d) plateau-like emission ($t^{-0.5}$) before a sharp powerlaw decline ($t^{-7}$). The plateau emission was in excess of $L_\textrm{X} \gtrsim 10^{46}$ ($10^{47}$) erg s$^{-1}$ at z>0.5 (2.0). Deep optical and radio observations resulted in non-detections of the transient. Our observations with Gemini South revealed a faint potential host galaxy ($r \approx 24$ AB mag) near the edge of the X-ray localization. The faint candidate host, and lack of other potential hosts ($r \gtrsim 26$ AB mag; $J \gtrsim 23$ AB mag), implies a higher redshift origin (z>0.5), which produces extreme X-ray properties that are inconsistent with many known extragalactic transient classes. In particular, the lack of a bright gamma-ray counterpart, with the isotropic-equivalent energy ($10 - 10,000$ keV) constrained by GECam and Konus-Wind to $E_{\gamma,\textrm{iso}} \lesssim 4\times10^{51}$ ($6\times10^{52}$) erg at z>0.5 (2.0), conflicts with known gamma-ray bursts (GRBs) of similar X-ray luminosities. We therefore favor a jetted tidal disruption event (TDE) as the progenitor of EP240408a at z>1.0, possibly caused by the disruption of a white dwarf by an intermediate mass black hole. The alternative is that EP240408a may represent a new, previously unknown class of transient., Comment: Accepted in ApJL. Minor revisions. Appendices from previous version combined with main text

We have analyzed the prompt and afterglow characteristics of the intermediate luminosity burst ``GRB 210210A". Our prompt emission analysis indicates that GRB 210210A is among the softest long GRBs detected by the Swift-BAT. The time-integrated prompt emission spectrum of GRB 210210A is aptly described by a power law function with an exponential cutoff. The spectral peak energy (E$_{p,z}$) in rest-frame and the E$_{\rm \gamma, iso}$ for this GRB marginally satisfy the 2$\sigma$ Amati correlation, a common feature observed in low/intermediate luminosity GRBs. Notably, an early bump is observed in the Swift-XRT light curve (a rare feature); the optical afterglow light curve, on the other hand, appears to follow a power law decay. However, due to the lack of sufficient early optical observations, we cannot completely rule out the possibility of an early bump in the optical light curve. For the bump observed in the early X-ray light curve, we calculated parameters such as peak time, rise time, decay time, and bulk Lorentz factor ($\Gamma_{0}$ $\sim$ 156), which perfectly satisfy the correlation between the parameters of the onset of the afterglow in GRBs. Both the optical and X-ray (including our observations) light curves exhibit a chromatic break in the late afterglow. Based on the prompt and afterglow parameters, we confirm that the intermediate luminosity GRB 210210A favors a collapsar scenario and is possibly powered by a magnetar., Comment: 17 pages, 9 figures, Revised version after addressing the referee's comments

Is there an algorithm to determine attributes such as positivity or non-zeroness of linear recurrence sequences? This long-standing question is known as Skolem's problem. In this paper, we study the complexity of an equivalent problem, namely the (generalized) moment membership problem for matrices. We show that this problem is decidable for orthogonal, unitary and real eigenvalue matrices, and undecidable for matrices over certain commutative and non-commutative polynomial rings. Our results imply that the positivity problem for simple unitary linear recurrence sequences is decidable, and is undecidable for linear recurrence sequences over the ring of commutative polynomials. As a byproduct, we prove a free version of Polya's theorem., Comment: 17 pages

Convex optimization encompasses a wide range of optimization problems, containing many efficiently solvable subclasses. Interior point methods are currently the state-of-the-art approach for solving such problems, particularly effective for classes like semidefinite programming, quadratic programming, and geometric programming. However, their success hinges on the construction of self-concordant barrier functions for the feasible sets. In this work, we introduce an alternative method for tackling convex optimization problems, employing a homotopy. With this technique, the feasible set of a trivial optimization problem is continuously transformed into the target one, while tracking the solutions. We conduct an analysis of this approach, focusing on its application to semidefinite programs, hyperbolic programs, and convex optimization problems with a single convexity constraint. Moreover, we demonstrate that our approach numerically outperforms state-of-the-art methods in several interesting cases., Comment: 29 pages, 10 figures

We use Markov categories to generalize the basic theory of Markov chains and hidden Markov models to an abstract setting. This comprises characterizations of hidden Markov models in terms of conditional independences and algorithms for Bayesian filtering and smoothing applicable in all Markov categories with conditionals. When instantiated in appropriate Markov categories, these algorithms specialize to existing ones such as the Kalman filter, forward-backward algorithm, and the Rauch-Tung-Striebel smoother. We also prove that the sequence of outputs of our abstract Bayes filter is itself a Markov chain with a concrete formula for its transition maps. There are two main features of this categorical framework. The first is its abstract generality, as manifested in our unified account of hidden Markov models and algorithms for filtering and smoothing in discrete probability, Gaussian probability, measure-theoretic probability, possibilistic nondeterminism and others at the same time. The second feature is the intuitive visual representation of information flow in terms of string diagrams., Comment: v3: shortened to 55 pages

We study when the Picard group of smooth surfaces of degree $d\geq 5$ in $\mathbb{P}^3$ acquires extra classes. In particular we show that the so called exceptional components of the Noether-Lefschetz locus are not Zariski dense. This answers a 1991 question of C. Voisin. We also obtain similar results for the Noether-Lefschetz locus for suitable $(Y,L)$, where $Y$ is a smooth projective threefold and $L$ a very ample line bundle. Both results are applications of the Zilber-Pink viewpoint recently developed by the authors for arbitrary (polarized, integral) variations of Hodge structures., Comment: To appear in IMRN

We obtained a 108 ks Chandra X-ray Observatory (CXO) observation of PSR J1849-0001 and its pulsar wind nebula (PWN), coincident with the TeV source HESS J1849-000. By analyzing the new and archival CXO data, we resolved the pulsar from the PWN, explored the PWN morphology on arcsecond and arcminute scales, and measured the spectra of different regions of the PWN. Both the pulsar and the compact inner PWN spectra are hard with power-law photon indices of $1.20 \pm 0.07$ and $1.49 \pm 0.20$, respectively. The jet-dominiated PWN has a relatively low luminosity, lack of gamma-ray pulsations, relatively hard and nonthermal spectrum of the pulsar, and sine-like pulse profile, which indicates a relatively small angle between the pulsar's spin and magnetic dipole axis. In this respect, it shares similar properties with a few other so-called MeV pulsars. Although the joint X-ray and TeV spectral energy distribution can be roughly described by a single-zone model, the obtained magnetic field value is unrealistically low. A more realistic scenario is the presence of a relic PWN, no longer emitting synchrotron X-rays but still radiating in TeV via inverse-Compton upscattering. We calso serendipitously detected surprisingly bright X-ray emission from a very wide binary whose components should not be interacting., Comment: 13 pages, 9 figures, Accepted by ApJ

Compact objects and supernova remnants provide nearby laboratories to probe the fate of stars after they die, and the way they impact, and are impacted by, their surrounding medium. The past five decades have significantly advanced our understanding of these objects, and showed that they are most relevant to our understanding of some of the most mysterious energetic events in the distant Universe, including Fast Radio Bursts and Gravitational Wave sources. However, many questions remain to be answered. These include: What powers the diversity of explosive phenomena across the electromagnetic spectrum? What are the mass and spin distributions of neutron stars and stellar mass black holes? How do interacting compact binaries with white dwarfs - the electromagnetic counterparts to gravitational wave LISA sources - form and behave? Which objects inhabit the faint end of the X-ray luminosity function? How do relativistic winds impact their surroundings? What do neutron star kicks reveal about fundamental physics and supernova explosions? How do supernova remnant shocks impact cosmic magnetism? This plethora of questions will be addressed with AXIS - the Advanced X-ray Imaging Satellite - a NASA Probe Mission Concept designed to be the premier high-angular resolution X-ray mission for the next decade. AXIS, thanks to its combined (a) unprecedented imaging resolution over its full field of view, (b) unprecedented sensitivity to faint objects due to its large effective area and low background, and (c) rapid response capability, will provide a giant leap in discovering and identifying populations of compact objects (isolated and binaries), particularly in crowded regions such as globular clusters and the Galactic Center, while addressing science questions and priorities of the US Decadal Survey for Astronomy and Astrophysics (Astro2020)., Comment: 61 pages, 33 figures. This White Paper is part of a series commissioned for the AXIS Probe Concept Mission

Latest since the landmark studies of Kovacs and co-workers on the glass transition of polymers, it is clear that thermally induced volume changes are of central importance for the understanding of the nature of the glass transition. Due to the kinetic background of the canonical (thermal) glass transition, it does not seem possible to derive a well-defined glass transition temperature T_g based on susceptibilities such as the thermal volume expansion coefficient, \beta(T), being strongly coupled to the glass transition process. Therefore, in practice, T_g is for example defined via the inflection point of the step-like \beta(T) curve. In this publication, we propose to use a thermo-optical feature, preceding the glass transition in the high-temperature phase, to determine the glass transition temperature T_g of a model polymer in a rather narrow temperature interval., Comment: Preprint

We report the discovery of Swift J221951-484240 (hereafter: J221951), a luminous slow-evolving blue transient that was detected by the Neil Gehrels Swift Observatory Ultra-violet/Optical Telescope (Swift/UVOT) during the follow-up of Gravitational Wave alert S190930t, to which it is unrelated. Swift/UVOT photometry shows the UV spectral energy distribution of the transient to be well modelled by a slowly shrinking black body with an approximately constant temperature of T~2.5x10^4 K. At a redshift z=0.5205, J221951 had a peak absolute magnitude of M_u,AB = -23 mag, peak bolometric luminosity L_max=1.1x10^45 erg s^-1 and a total radiated energy of E>2.6x10^52 erg. The archival WISE IR photometry shows a slow rise prior to a peak near the discovery date. Spectroscopic UV observations display broad absorption lines in N V and O VI, pointing toward an outflow at coronal temperatures. The lack of emission in the higher H~Lyman lines, N I and other neutral lines is consistent with a viewing angle close to the plane of the accretion or debris disc. The origin of J221951 can not be determined with certainty but has properties consistent with a tidal disruption event and the turn-on of an active galactic nucleus., Comment: 37 pages (25 main + 12 supplementary), submitted to MNRAS

The matrix rank and its positive versions are robust for small approximations, i.e. they do not decrease under small perturbations. In contrast, the multipartite tensor rank can collapse for arbitrarily small errors, i.e. there may be a gap between rank and border rank, leading to instabilities in the optimization over sets with fixed tensor rank. Can multipartite positive ranks also collapse for small perturbations? In this work, we prove that multipartite positive and invariant tensor decompositions exhibit gaps between rank and border rank, including tensor rank purifications and cyclic separable decompositions. We also prove a correspondence between positive decompositions and membership in certain sets of multipartite probability distributions, and leverage the gaps between rank and border rank to prove that these correlation sets are not closed. It follows that testing membership of probability distributions arising from resources like translational invariant Matrix Product States is impossible in finite time. Overall, this work sheds light on the instability of ranks and the unique behavior of bipartite systems., Comment: 14 pages, 21 pages of appendices, 12 figures. v3: published version

We report the discovery of the unusually bright long-duration gamma-ray burst (GRB), GRB 221009A, as observed by the Neil Gehrels Swift Observatory (Swift), Monitor of All-sky X-ray Image (MAXI), and Neutron Star Interior Composition Explorer Mission (NICER). This energetic GRB was located relatively nearby (z = 0.151), allowing for sustained observations of the afterglow. The large X-ray luminosity and low Galactic latitude (b = 4.3 degrees) make GRB 221009A a powerful probe of dust in the Milky Way. Using echo tomography we map the line-of-sight dust distribution and find evidence for significant column densities at large distances (~> 10kpc). We present analysis of the light curves and spectra at X-ray and UV/optical wavelengths, and find that the X-ray afterglow of GRB 221009A is more than an order of magnitude brighter at T0 + 4.5 ks than any previous GRB observed by Swift. In its rest frame GRB 221009A is at the high end of the afterglow luminosity distribution, but not uniquely so. In a simulation of randomly generated bursts, only 1 in 10^4 long GRBs were as energetic as GRB 221009A; such a large E_gamma,iso implies a narrow jet structure, but the afterglow light curve is inconsistent with simple top-hat jet models. Using the sample of Swift GRBs with redshifts, we estimate that GRBs as energetic and nearby as GRB 221009A occur at a rate of ~<1 per 1000 yr - making this a truly remarkable opportunity unlikely to be repeated in our lifetime., Comment: 30 pages, 13 figures, submitted to ApJL

PSR J1101-6101 is an energetic young pulsar which powers the remarkable Lighthouse pulsar wind nebula (PWN). The pulsar belongs to the rare type of radio- and gamma-ray-quiet pulsars which are bright in hard X-rays. Moreover, the Lighthouse PWN is remarkable for its misaligned outflow (which gave rise to the PWN's nickname). Also known as "pulsar filaments", these collimated parsec-scale X-ray structures have been recently discovered in the vicinity of a handful of fast-moving pulsars, and appear unaffected by the ram pressure which confines pulsar tails. We report on NuSTAR observations of PSR J1101-6101 and its misaligned outflow -- the first observation of such a structure above ~10 keV. We detect the outflow up to 25 keV, spatially resolve its spectral evolution with distance from the pulsar, find unambiguous evidence of spectral cooling with distance from the pulsar, and infer physical properties of the particles and magnetic field in the outflow. We also reanalzye archival Chandra data and discuss the outflow's small-scale structure. We detect pulsations from PSR J1101-6101 up to 20 keV, present the X-ray pulse profile, confirm its period derivative, and perform phase-resolved spectroscopy. Lastly, we discuss the X-ray source 2CXO J110158.4-605649 = 2XMM J110158.5--605651 (a serendipitously observed blazar) and suggest it may be the X-ray counterpart to the GeV source 4FGL J1102.0--6054., Comment: 16 pages, 9 figures, ApJ submitted

Several physically inspired problems have been proven undecidable; examples are the spectral gap problem and the membership problem for quantum correlations. Most of these results rely on reductions from a handful of undecidable problems, such as the halting problem, the tiling problem, the Post correspondence problem or the matrix mortality problem. All these problems have a common property: they have an NP-hard bounded version. This work establishes a relation between undecidable unbounded problems and their bounded NP-hard versions. Specifically, we show that NP-hardness of a bounded version follows easily from the reduction of the unbounded problems. This leads to new and simpler proofs of the NP-hardness of bounded version of the Post correspondence problem, the matrix mortality problem, the positivity of matrix product operators, the reachability problem, the tiling problem, and the ground state energy problem. This work sheds light on the intractability of problems in theoretical physics and on the computational consequences of bounding a parameter., Comment: 10 pages and 7 pages of appendices, 8 figures; v2,v3: minor changes

We have collected a new deep {\it Chandra X-ray Observatory} ({\it CXO}) exposure of PSR B2224+65 and the `Guitar Nebula', mapping the complex X-ray structure. This is accompanied by a new {\it HST} H$\alpha$ image of the head of the Guitar. Comparing the {\it HST} and {\it CXO} structures in 4 epochs over 25 years, we constrain the evolution of the TeV particles that light up the filament. Cross-field diffusion appears to be enhanced, likely by the injected particles, behind the filament's sharp leading edge, explaining the filament width and its evolving surface brightness profile., Comment: 13 pages, 10 figures, accepted for publication in ApJ