Search

This document presents the solutions to the exercises in the book "Albert algebras over commutative rings" published by Cambridge University Press, 2024., Comment: Approximately 300 pages

Over recent decades, astronomy has entered the era of massive data and real-time surveys. This is improving the study of transient objects - although they still contain some of the most poorly understood phenomena in astrophysics, as it is inherently more difficult to obtain data on them. In order to help detect these objects in their brightest state, we have built a quasi-real time transient detection system for the XMM-Newton pipeline: the Search for Transient Objects in New detections using Known Sources (STONKS) pipeline. STONKS detects long-term X-ray transients by automatically comparing new XMM-Newton detections to any available archival X-ray data at this position, sending out an alert if the amplitude of variability between observations is over 5. This required an initial careful cross-correlation and flux calibration of various X-ray catalogs from different observatories (XMM-Newton, Chandra, Swift, ROSAT, and eROSITA). We also systematically computed the XMM-Newton upper limits at the position of any X-ray source covered by the XMM-Newton observational footprint, even without any XMM-Newton counterpart. The behavior of STONKS was then tested on all 483 observations performed with imaging mode in 2021. Over the 2021 testing run, STONKS provided $0.7^{+0.7}_{-0.5}$ alerts per day, about 80% of them being serendipitous. STONKS also detected targeted tidal disruption events, ensuring its ability to detect other serendipitous events. As a byproduct of our method, the archival multi-instrument catalog contains about one million X-ray sources, with 15% of them involving several catalogs and 60% of them having XMM-Newton upper limits. STONKS demonstrates a great potential for revealing future serendipitous transient X-ray sources, providing the community with the ability to follow-up on these objects a few days after their detection., Comment: 30 pages, 30 figures, accepted in A&A

We give a generalization of Lagarias' formula for diffraction by ideal crystals, and we apply it to the lattice case, in preparation for addressing the problem of quasicrystals and complex dimensions posed by Lapidus and van Frankenhuijsen concerning the quasiperiodic properties of the set of complex dimensions of any nonlattice self-similar fractal string. More specifically, in this paper, we consider the case of the complex dimensions of a lattice (rather than of a nonlattice) self-similar string and show that the corresponding diffraction measure exists, is unique, and is given by a suitable $\textit{continuous}$ analogue of a discrete Dirac comb. We also obtain more general results concerning the autocorrelation measures and diffraction measures of generalized idealized fractals associated to possibly degenerate lattices and the corresponding extension of the Poisson Summation Formula., Comment: A new version with a few updates, 27 pages, 4 figures. arXiv admin note: text overlap with arXiv:2009.03493

Albert algebras, a specific kind of Jordan algebra, are naturally distinguished objects among commutative non-associative algebras and also arise naturally in the context of simple affine group schemes of type $F_4$, $E_6$, or $E_7$. We study these objects over an arbitrary base ring $R$, with particular attention to the case of the integers. We prove in this generality results previously in the literature in the special case where $R$ is a field of characteristic different from 2 and 3., Comment: v2: section 12 on number of generators is new, Theorem 13.5 now holds for semi-local rings (and even a somewhat wider class)

The local theory of complex dimensions for real and $p$-adic fractal strings describes oscillations that are intrinsic to the geometry, dynamics and spectrum of archimedean and nonarchimedean fractal strings. We aim to develop a global theory of complex dimensions for ad\`elic fractal strings in order to reveal the oscillatory nature of ad\`elic fractal strings and to understand the Riemann hypothesis in terms of the vibrations and resonances of fractal strings. We present a simple and natural construction of self-similar $p$-adic fractal strings of any rational dimension in the closed unit interval $[0,1]$. Moreover, as a first step towards a global theory of complex dimensions for ad\`elic fractal strings, we construct an ad\`elic Cantor string in the set of finite ad\`eles $\mathbb{A}_0$ as an infinite Cartesian product of every $p$-adic Cantor string, as well as an ad\`elic Cantor-Smith string in the ring of ad\`eles $\mathbb{A}$ as a Cartesian product of the general Cantor string and the ad\`elic Cantor string., Comment: 16 pages

Noncommutative geometry provides a framework, via the construction of spectral triples, for the study of the geometry of certain classes of fractals. Many fractals are constructed as natural limits of certain sets with a simpler structure: for instance, the Sierpi\'nski is the limit of finite graphs consisting of various affine images of an equilateral triangle. It is thus natural to ask whether the spectral triples, constructed on a class of fractals called piecewise $C^1$-fractal curves, are indeed limits, in an appropriate sense, of spectral triples on the approximating sets. We answer this question affirmatively in this paper, where we use the spectral propinquity on the class of metric spectral triples, in order to formalize the sought-after convergence of spectral triples. Our results and methods are relevant to the study of analysis on fractals and have potential physical applications., Comment: 35 pages, 3 figures. To appear in Adv. Math

The Lattice String Approximation algorithm (or LSA algorithm) of M. L. Lapidus and M. van Frankenhuijsen is a procedure that approximates the complex dimensions of a nonlattice self-similar fractal string by the complex dimensions of a lattice self-similar fractal string. The implication of this procedure is that the set of complex dimensions of a nonlattice string has a quasiperiodic pattern. Using the LSA algorithm, together with the multiprecision polynomial solver MPSolve which is due to D. A. Bini, G. Fiorentino and L. Robol, we give a new and significantly more powerful presentation of the quasiperiodic patterns of the sets of complex dimensions of nonlattice self-similar fractal strings. The implementation of this algorithm requires a practical method for generating simultaneous Diophantine approximations, which in some cases we can accomplish by the continued fraction process. Otherwise, as was suggested by Lapidus and van Frankenhuijsen, we use the LLL algorithm of A. K. Lenstra, H. W. Lenstra, and L. Lov\'asz., Comment: 38 pages, 11 figures, 7 tables

The XMM-Newton Survey Science Centre Consortium (SSC) develops software in close collaboration with the Science Operations Centre to perform a pipeline analysis of all XMM-Newton observations. In celebration of the 20th launch anniversary, the SSC has compiled the 4th generation of serendipitous source catalogues, 4XMM. The catalogue described here, 4XMM-DR9s, explores sky areas that were observed more than once by XMM-Newton. It was constructed from simultaneous source detection on the overlapping observations, which were bundled in groups ("stacks"). Stacking leads to a higher sensitivity, resulting in newly discovered sources and better constrained source parameters, and unveils long-term brightness variations. As a novel feature, positional rectification was applied beforehand. Observations with all filters and suitable camera settings were included. Exposures with a high background were discarded, which was determined through a statistical analysis of all exposures in each instrument configuration. The X-ray background maps used in source detection were modelled via adaptive smoothing with newly determined parameters. Source fluxes were derived for all contributing observations, irrespective of whether the source would be detectable in an individual observation. From 1,329 stacks with 6,604 contributing observations over repeatedly covered 300 square degrees in the sky, 4XMM-DR9s lists 288,191 sources. 218,283 of them were observed several times. Most stacks are composed of two observations, the largest one comprises 352. The number of observations of a source ranges from 1 to 40. Auxiliary products like X-ray images, long-term light curves, and optical finding charts are published as well. 4XMM-DR9s is considered a prime resource to explore long-term variability of X-ray sources discovered by XMM-Newton. Regular incremental releases including new public observations are planned., Comment: Accepted for publication in A&A. 20 pages

Sky surveys produce enormous quantities of data on extensive regions of the sky. The easiest way to access this information is through catalogues of standardised data products. {\em XMM-Newton} has been surveying the sky in the X-ray, ultra-violet, and optical bands for 20 years. The {\em XMM-Newton} Survey Science Centre has been producing standardised data products and catalogues to facilitate access to the serendipitous X-ray sky. Using improved calibration and enhanced software, we re-reduced all of the 14041 {\em XMM-Newton} X-ray observations, of which 11204 observations contained data with at least one detection and with these we created a new, high quality version of the {\em XMM-Newton} serendipitous source catalogue, 4XMM-DR9. 4XMM-DR9 contains 810795 detections down to a detection significance of 3 $\sigma$, of which 550124 are unique sources, which cover 1152 degrees$^{2}$ (2.85\%) of the sky. Filtering 4XMM-DR9 to retain only the cleanest sources with at least a 5 $\sigma$ detection significance leaves 433612 detections. Of these detections, 99.6\% have no pileup. Furthermore, 336 columns of information on each detection are provided, along with images. The quality of the source detection is shown to have improved significantly with respect to previous versions of the catalogues. Spectra and lightcurves are also made available for more than 288000 of the brightest sources (36\% of all detections)., Comment: 14 pages, 11 figures, accepted to be published Astronomy and Astrophysics (A&A)

Ferropericlase (Fp), (Mg$_\mathrm{1-x}$Fe$_\mathrm{x}$)O, is the second most abundant phase in the Earths lower mantle. At relevant pressure-temperature conditions, iron in Fp undergoes a high spin (HS), S=2, to low spin (LS), S=0, state change. The nature of this phenomenon is quite well understood now, but there are still basic questions regarding the structural stability and the existence of soft phonon modes during this iron state change. General theories exist to explain the volume reduction, the significant thermo-elastic anomalies, and the broad nature of this HS-LS crossover. These theories make extensive use of the quasi-harmonic approximation (QHA). Therefore, dynamical and structural stability is essential to their validity. Here, we investigate the vibrational spectrum of Fp throughout this spin-crossover using $\textit{ab initio}$ DFT+Usc calculations. We address vibrational modes associated with isolated and (2nd) nearest neighbor iron ions undergoing the HS-LS state change. As expected, acoustic modes of this solid solution are resilient while optical modes are the most affected. We show that there are no soft phonon modes across this HS-LS crossover, and Fp is dynamically stable at all relevant pressures.

We study the essential singularities of geometric zeta functions $\zeta_{\mathcal L}$, associated with bounded fractal strings $\mathcal L$. For any three prescribed real numbers $D_{\infty}$, $D_1$ and $D$ in $[0,1]$, such that $D_{\infty}\alpha\}$, except for possible isolated singularities in this half-plane. Defining $\mathcal L$ as the disjoint union of a sequence of suitable generalized Cantor strings, we show that the set of accumulation points of the set $S_{\infty}$ of essential singularities of $\zeta_{\mathcal L}$, contained in the open right half-plane $\{{\rm Re}\, s>D_{\infty}\}$, coincides with the vertical line $\{{\rm Re}\, s=D_{\infty}\}$. We extend this construction to the case of distance zeta functions $\zeta_A$ of compact sets $A$ in $\mathbb{R}^N$, for any positive integer $N$., Comment: Theorem 3.2 (b) was wrong in the previous version, so we have decided to omit it and pursue this issue at some future time. Part (b) of Theorem 3.2. was not used anywhere else in the paper. Theorem 3.2. is now called Proposition 3.2. on page 12. Corrected minor typos and added new references To appear in: Pure and Applied Functional Analysis; issue 5 of volume 5 (2020)

Model-based reinforcement learning is an appealing framework for creating agents that learn, plan, and act in sequential environments. Model-based algorithms typically involve learning a transition model that takes a state and an action and outputs the next state---a one-step model. This model can be composed with itself to enable predicting multiple steps into the future, but one-step prediction errors can get magnified, leading to unacceptable inaccuracy. This compounding-error problem plagues planning and undermines model-based reinforcement learning. In this paper, we address the compounding-error problem by introducing a multi-step model that directly outputs the outcome of executing a sequence of actions. Novel theoretical and empirical results indicate that the multi-step model is more conducive to efficient value-function estimation, and it yields better action selection compared to the one-step model. These results make a strong case for using multi-step models in the context of model-based reinforcement learning.

We present observations of the possible short GRB 180418A in $\gamma$-rays, X-rays, and in the optical. Early optical photometry with the TAROT and RATIR instruments show a bright peak ($\approx$ 14.2 AB mag) between $T+28$ and $T+90$ seconds that we interpret as the signature of a reversal shock. Later observations can be modeled by a standard forward shock model and show no evidence of jet break, allowing us to constrain the jet collimation to $\theta_j> 7^\circ$. Using deep late-time optical observations we place an upper limit of $r>24$ AB mag on any underlying host galaxy. The detection of the afterglow in the \textit{Swift} UV filters constrains the GRB redshift to $z<1.3$ and places an upper bound on the $\gamma$-ray isotropic equivalent energy $E_{\rm{\gamma,iso}} < 3 \times 10^{51}$ erg. The properties of this GRB (e.g. duration, hardness ratio, energetic, and environment) lie at the intersection between short and long bursts, and we can not conclusively identify its type. We estimate that the probability that it is drawn from the population of short GRBs is 10\%-30\%., Comment: Accepted por publication in ApJ

XMM-Newton has observed the X-ray sky since early 2000. The XMM-Newton Survey Science Centre Consortium has published catalogues of X-ray and ultraviolet sources found serendipitously in the individual observations. This series is now augmented by a catalogue dedicated to X-ray sources detected in spatially overlapping XMM-Newton observations. The aim of this catalogue is to explore repeatedly observed sky regions. It thus makes use of the long(er) effective exposure time per sky area and offers the opportunity to investigate long-term flux variability directly through the source detection process. A new standardised strategy for simultaneous source detection on multiple observations is introduced. It is coded as a new task within the XMM-Newton Science Analysis System and used to compile a catalogue of sources from 434 stacks comprising 1,789 overlapping XMM-Newton observations that entered the 3XMM-DR7 catalogue, have a low background and full-frame readout of all EPIC cameras. The first stacked catalogue is called 3XMM-DR7s. It contains 71,951 unique sources with positions and parameters such as fluxes, hardness ratios, quality estimates, and information on inter-observation variability. About 15% of the sources are new with respect to 3XMM-DR7. Through stacked source detection, the parameters of repeatedly observed sources can be determined with higher accuracy than in the individual observations. The method is more sensitive to faint sources and tends to produce fewer spurious detections. With this first stacked catalogue we demonstrate the feasibility and benefit of the approach. It supplements the large data base of XMM-Newton detections by additional, in particular faint, sources and adds variability information. In the future, the catalogue will be expanded to larger samples and continued within the series of serendipitous XMM-Newton source catalogues., Comment: Accepted for publication in A&A. 27 pages, ~4MB

We present a theoretical investigation, based on ab initio calculations and the quasi-harmonic approximation, on the stability properties of magnesium (MgCO$_3$) and calcium (CaCO$_3$) carbonates at high temperatures and pressures. The results indicate that those carbonates should be stable in Earth's lower mantle, instead of dissociating into other minerals, in chemical environments with excesses of SiO$_2$, MgO, or MgSiO$_3$. Therefore, considering the lower mantle chemical composition, filled with major minerals such as MgSiO$_3$ and MgO, calcium and magnesium carbonates are the primary candidates as carbon hosts in that region. For the thermodynamic conditions of the mantle, the results also indicate that carbon should be primarily hosted on MgCO$_3$. Finally, the results indicate that carbon, in the form of free CO$_2$, is unlikely in the lower mantle.

Our main goal in this long survey article is to provide an overview of the theory of complex fractal dimensions and of the associated geometric or fractal zeta functions, first in the case of fractal strings (one-dimensional drums with fractal boundary), in \S2, and then in the higher-dimensional case of relative fractal drums and, in particular, of arbitrary bounded subsets of Euclidean space of $\mbr^N$, for any integer $N \geq 1$, in \S3. Special attention is paid to discussing a variety of examples illustrating the general theory rather than to providing complete statements of the results and their proofs, for which we refer to the author's previous (joint) books mentioned in the paper. Finally, in an epilogue (\S4), entitled "From quantized number theory to fractal cohomology", we briefly survey aspects of related work (motivated in part by the theory of complex fractal dimensions) of the author with H. Herichi (in the real case) [HerLap1], along with [Lap8], and with T. Cobler (in the complex case) [CobLap1], respectively, as well as in the latter part of a book in preparation by the author, [Lap10]., Comment: To appear in: {\em Horizons of Fractal Geometry and Complex Dimensions} (R.~G. Niemeyer, E.~P.~J. Pearse, J.~A. Rock and T. Samuel, eds.), Contemporary Mathematics, Amer. Math. Soc., Providence, R.~I., 2019

Thermal equations of state (EoS) are essential in several scientific domains. However, experimental determination of EoS parameters may be limited at extreme conditions, therefore, {\it ab~initio} calculations have become an important method to obtain them. Density Functional Theory (DFT) and its extensions with various degrees of approximations for the exchange and correlation (XC) energy is the method of choice, but large errors in the EoS parameters are still common. The alkali halides have been problematic from the onset of this field and the quest for appropriate DFT functionals for such ionic and relatively weakly bonded systems has remained an active topic of research. Here we use DFT + van der Waals functionals to calculate vibrational properties, thermal EoS, thermodynamic properties, and the B1 to B2 phase boundary of NaCl. Our results reveal i) a remarkable improvement over the performance of standard Local Density Approximation and Generalized Gradient Approximation functionals for all these properties and phase transition boundary, as well as ii) great sensitivity of anharmonic effects on the choice of XC functional.