Search

Let $K$ be an imaginary quadratic field in which the odd prime $p$ does not split. When the $p$-part of the class group of $K$ is cyclic, we describe the possible structures for the $p$-part of the class group of the first level of the cyclotomic $\mathbb{Z}_p$-extension of $K$. This allows us to show the compatibility of the heuristics of Cohen--Lenstra--Martinet for class groups with the heuristics of Ellenberg--Jain--Venkatesh for how often the cyclotomic Iwasawa invariant $\lambda$ equals 1.

This paper addresses, for the first time, a key aspect of the phenomenology of Compact Symmetric Objects (CSOs) -- the characteristics of their radio spectra. We present a radio-spectrum description of a complete sample of high luminosity CSOs (CSO-2s), which shows that they exhibit the \textit{complete} range of spectral types, including flat-spectrum sources ($\alpha \ge -0.5$), steep-spectrum sources ($\alpha < -0.5$), and peaked-spectrum sources. We show that there is no clear correlation between spectral type and size, but there is a correlation between the high-frequency spectral index and both object type and size. We also show that, to avoid biasing the data and to understand the various classes of jetted-AGN involved, the complete range of spectral types should be included in studying the general phenomenology of compact jetted-AGN, and that complete samples must be used, selected over a wide range of frequencies. We discuss examples that demonstrate these points. We find that the high-frequency spectral indices of CSO-2s span $-1.3 <\alpha_{\rm hi} < -0.3$, and hence that radio spectral signatures cannot be used to discriminate definitively between CSO-2s, binary galactic nuclei, and millilensed objects, unless they have $\alpha_{\rm hi} >-0.3$.

Owens Valley Radio Observatory (OVRO) observations of supermassive black hole binary (SMBHB) candidate PKS~2131$-$021 revealed, for the first time, six likely characteristics of the phenomenology exhibited by SMBHB in blazars, of which the most unexpected and critical is sinusoidal flux density variations. We have now identified a second blazar, PKS~J0805$-$0111, showing significant sinusoidal variations, with an observed period that translates to $1.422 \pm 0.005$ yr in the rest frame of the $z = 1.388$ object. We generate $10^6$ simulated light curves to reproduce the radio variability characteristics of PKS~J0805$-$0111, and show that the global probability, considering the \textit{look-elsewhere effect}, indicates that the observed periodicity can be attributed to the red noise tail of the power spectral density, with a $p_0$ value of $7.8 \times 10^{-5}$ (i.e. 3.78$\sigma$). PKS J0805$-$0111 displays all six characteristics observed in PKS 2131$-$021. Taking into account the well-defined OVRO sample size, the false positive probability $\sim 0.22$, but the rare behavior makes this a strong SMBHB candidate. The discovery of a second SMBHB candidate exhibiting these rare characteristics reveals that PKS~2131$-$021 is not a unique, isolated case. With these two strong cases we are clearly seeing only the tip of the iceberg. We estimate that the number of SMBHB candidates amongst blazars $\sim$ 1 in 100.

Haystack and Owens Valley Radio Observatory (OVRO) observations recently revealed strong sinusoidal total flux density variations that maintained coherence between 1975 and 2021 in the blazar PKS 2131-021 ($z=1.283)$. This was interpreted as possible evidence of a supermassive black hole binary (SMBHB). Extended observations through 2023 show coherence over 47.9~years, with an observed period $P_\textrm{15 GHz}=(1739.3 \pm 1.2) \, {\rm days}$. We reject, with $p$-value = $5.3 \times 10^{-7}$, the hypothesis that the variations are due to random fluctuations in the red noise tail of the power spectral density. There is clearly a constant-period physical phenomenon in PKS 2131-021 producing coherent intermittent sinusoidal flux density variations. We find the coherent sinusoidal intensity variations extend from below 2.7 GHz to optical frequencies, from which we derive an observed period $P_\textrm{optical}=(1764 \pm 36)$ days. Across this broad frequency range there is a monotonic phase shift in the sinusoidal variations with frequency. The same coherent periodicity is possibly also observed at $\gamma$-ray energies. The importance of well-vetted SMBHB candidates to searches for gravitational waves is pointed out. We estimate the fraction of blazars that are SMBHB candidates to be $>1$ in 100. Thus monitoring programs covering tens of thousands of blazars could discover hundreds of SMBHB candidates., Comment: 23 pages, 17 figures, 5 tables, submitted to ApJ

Deciding whether or not two polynomials have isomoprhic splitting fields over the rationals is the Field Isomorphism Problem. We consider polynomials of the form $f_n(x) = x^4-nx^3-6x^2+nx+1$ with $n \neq 3$ a positive integer and we let $K_n$ denote the splitting field of $f_n(x)$; a `simplest quartic field'. Our main theorem states that under certain hypotheses there can be at most one positive integer $m \neq n$ such that $K_m=K_n$. The proof relies on the existence of squares in recurrent sequences and a result of J.H.E. Cohn [3]. These sequences allow us to establish uniqueness of the splitting field under additional hypotheses in Section (5) and to establish a connection with elliptic curves in Section (6)., Comment: 10 pages

The Carbon monOxide Mapping Array Project (COMAP) Pathfinder survey continues to demonstrate the feasibility of line-intensity mapping using high-redshift carbon monoxide (CO) line emission traced at cosmological scales. The latest COMAP Pathfinder power spectrum analysis is based on observations through the end of Season 2, covering the first three years of Pathfinder operations. We use our latest constraints on the CO(1-0) line-intensity power spectrum at $z\sim3$ to update corresponding constraints on the cosmological clustering of CO line emission and thus the cosmic molecular gas content at a key epoch of galaxy assembly. We first mirror the COMAP Early Science interpretation, considering how Season 2 results translate to limits on the shot noise power of CO fluctuations and the bias of CO emission as a tracer of the underlying dark matter distribution. The COMAP Season 2 results place the most stringent limits on the CO tracer bias to date, at $\langle{Tb}\rangle<4.8$ $\mu$K. These limits narrow the model space significantly compared to previous CO line-intensity mapping results while maintaining consistency with small-volume interferometric surveys of resolved line candidates. The results also express a weak preference for CO emission models used to guide fiducial forecasts from COMAP Early Science, including our data-driven priors. We also consider directly constraining a model of the halo-CO connection, and show qualitative hints of capturing the total contribution of faint CO emitters through the improved sensitivity of COMAP data. With continued observations and matching improvements in analysis, the COMAP Pathfinder remains on track for a detection of cosmological clustering of CO emission., Comment: 9 pages + bibliography and appendices (13 pages total); 9 figures, 1 table; v2 reflects minor changes made for version submitted to A&A, with no changes to top-line results

We present updated constraints on the cosmological 3D power spectrum of carbon monoxide CO(1-0) emission in the redshift range $2.4$-$3.4$. The constraints are derived from the two first seasons of Carbon monOxide Mapping Array Project (COMAP) Pathfinder line-intensity mapping observations aiming to trace star-formation during the Epoch of Galaxy Assembly. These results improve on the previous Early Science (ES) results through both increased data volume and improved data processing methodology. On the methodological side, we now perform cross-correlations between groups of detectors (''feed-groups''), as opposed to cross-correlations between single feeds, and this new feed-group pseudo power spectrum (FGPXS) is constructed to be more robust against systematic effects. In terms of data volume, the effective mapping speed is significantly increased due to an improved observational strategy as well as better data selection methodology. The updated spherically- and field-averaged FGPXS, $\tilde{C}(k)$, is consistent with zero, at a probability-to-exceed of around $34\,\%$, with an excess of $2.7\,\sigma$ in the most sensitive bin. Our power spectrum estimate is about an order of magnitude more sensitive in our six deepest bins across ${0.09\,\mathrm{Mpc}^{-1} < k < 0.73\,\mathrm{Mpc}^{-1}}$, as compared to the feed-feed pseudo power spectrum (FPXS) of COMAP ES. Each of these bins individually constrains the CO power spectrum to ${kP_\mathrm{CO}(k)< 2400-4900\,\mathrm{\mu K^2 Mpc^{2}}}$ at $95\,\%$ confidence. To monitor potential contamination from residual systematic effects, we analyze a set of 312 difference-map null tests and find that these are consistent with the instrumental noise prediction. In sum, these results provide the strongest direct constraints on the cosmological 3D CO(1-0) power spectrum published to date., Comment: 17 pages, 10 figures, v2 reflects changes made for version submitted to Astronomy and Astrophysics, addition of additional figure clarifying methodology, no change to final results

The CO Mapping Array Project (COMAP) Pathfinder is performing line intensity mapping of CO emission to trace the distribution of unresolved galaxies at redshift $z \sim 3$. We present an improved version of the COMAP data processing pipeline and apply this to the first two seasons of observations. This analysis improves on the COMAP Early Science (ES) results in several key aspects. On the observational side, all second season scans were made in constant-elevation mode, after noting that the previous Lissajous scans were associated with increased systematic errors; those scans accounted for 50% of the total Season 1 data volume. Secondly, all new observations were restricted to an elevation range of 35-65 degrees, to minimize sidelobe ground pickup. On the data processing side, more effective data cleaning in both the time- and map-domain has allowed us to eliminate all data-driven power spectrum-based cuts. This increases the overall data retention and reduces the risk of signal subtraction bias. On the other hand, due to the increased sensitivity, two new pointing-correlated systematic errors have emerged, and we introduce a new map-domain PCA filter to suppress these. Subtracting only 5 out of 256 PCA modes, we find that the standard deviation of the cleaned maps decreases by 67% on large angular scales, and after applying this filter, the maps appear consistent with instrumental noise. Combining all these improvements, we find that each hour of raw Season 2 observations yields on average 3.2 times more cleaned data compared to ES analysis. Combining this with the increase in raw observational hours, the effective amount of data available for high-level analysis is a factor of 8 higher than in ES. The resulting maps have reached an uncertainty of $25$-$50\,\mu K$ per voxel, providing by far the strongest constraints on cosmological CO line emission published to date., Comment: 23 pages, 22 figures, for submission to Astronomy and Astrophysics

The COVID-19 pandemic has highlighted the need to upgrade systems for infectious disease surveillance and forecasting and modeling of the spread of infection, both of which inform evidence-based public health guidance and policies. Here, we discuss requirements for an effective surveillance system to support decision making during a pandemic, drawing on the lessons of COVID-19 in the U.S., while looking to jurisdictions in the U.S. and beyond to learn lessons about the value of specific data types. In this report, we define the range of decisions for which surveillance data are required, the data elements needed to inform these decisions and to calibrate inputs and outputs of transmission-dynamic models, and the types of data needed to inform decisions by state, territorial, local, and tribal health authorities. We define actions needed to ensure that such data will be available and consider the contribution of such efforts to improving health equity.

Formalised libraries of combinatorial mathematics have rapidly expanded over the last five years, but few use one of the most important tools: probability. How can often intuitive probabilistic arguments on the existence of combinatorial structures, such as hypergraphs, be translated into a formal text? We present a modular framework using locales in Isabelle/HOL to formalise such probabilistic proofs, including the basic existence method and first formalisation of the Lov\'asz local lemma, a fundamental result in probability. The formalisation focuses on general, reusable formal probabilistic lemmas for combinatorial structures, and highlights several notable gaps in typical intuitive probabilistic reasoning on paper. The applicability of the techniques is demonstrated through the formalisation of several classic lemmas on the existence of hypergraphs with certain colourings., Comment: v2: updated to reflect anonymous reviewer feedback. Accepted to CPP2024

We present constraints on cosmological parameters using maps from the last Planck data release (PR4). In particular, we detail an upgraded version of the cosmic microwave background likelihood, HiLLiPoP, based on angular power spectra and relying on a physical modelling of the foreground residuals in the spectral domain. This new version of the likelihood retains a larger sky fraction (up to 75%) and uses an extended multipole range. Using this likelihood, along with low-l measurements from LoLLiPoP, we derive constraints on $\Lambda$CDM parameters that are in good agreement with previous Planck 2018 results, but with 10% to 20% smaller uncertainties. We demonstrate that the foregrounds can be accurately described in spectra domain with only negligible impact on $\Lambda$CDM parameters. We also derive constraints on single-parameter extensions to $\Lambda$CDM including $A_L$, $\Omega_K$, $N_{eff}$, and $\sum m_{\nu}$. Noteworthy results from this updated analysis include a lensing amplitude value of $A_L = 1.039 \pm 0.052$, which aligns more closely with theoretical expectations within the $\Lambda$CDM framework. Additionally, our curvature measurement, $\Omega_K = -0.012 \pm 0.010$, now demonstrates complete consistency with a flat universe, and our measurement of $S_8$ is closer to the measurements derived from large-scale structure surveys (at the 1.6$\sigma$ level). We also add constraints from PR4 lensing, making the combination the most constraining data set that is currently available from Planck. Additionally we explore adding baryon acoustic oscillation data, which tightens limits on some particular extensions to the standard cosmology., Comment: 18 pages, 23 figures, 2 appendix pages. Add constraints with BAO and lensing

ALEXANDRIA is an ERC-funded project that started in 2017, with the aim of bringing formal verification to mathematics. The past six years have seen great strides in the formalisation of mathematics and also in some relevant technologies, above all machine learning. Six years of intensive formalisation activity seem to show that even the most advanced results, drawing on multiple fields of mathematics, can be formalised using the tools available today., Comment: Invited paper for CICM 2023 (Conference on Intelligent Computer Mathematics). This revised version adds two references

Twenty-six years ago a small committee report, building on earlier studies, expounded a compelling and poetic vision for the future of astronomy, calling for an infrared-optimized space telescope with an aperture of at least $4m$. With the support of their governments in the US, Europe, and Canada, 20,000 people realized that vision as the $6.5m$ James Webb Space Telescope. A generation of astronomers will celebrate their accomplishments for the life of the mission, potentially as long as 20 years, and beyond. This report and the scientific discoveries that follow are extended thank-you notes to the 20,000 team members. The telescope is working perfectly, with much better image quality than expected. In this and accompanying papers, we give a brief history, describe the observatory, outline its objectives and current observing program, and discuss the inventions and people who made it possible. We cite detailed reports on the design and the measured performance on orbit., Comment: Accepted by PASP for the special issue on The James Webb Space Telescope Overview, 29 pages, 4 figures

A detailed knowledge of the dielectric properties of food materials is vital to any electromagnetic-based treatments, ranging from reheating meals in a domestic microwave oven through to sterilization processes. The uniformity and rate of heating of the food is highly dependent upon the dielectric constant and dielectric loss of the food material, with both parameters being intrinsically temperature-dependent. In this work, we explore various methods for the dielectric characterisation of materials over a wide frequency range (1 MHz to 20 GHz), to cover RF and microwave frequencies, as well as understand the impact of ingredients (e.g. salt, sugar) and temperature on free and bound water resonances. Such a large frequency range gives information that is invaluable when designing future equipment that relies on multi-frequency/broadband microwave heating techniques. The characterization methods of interest are the open-ended coaxial dielectric probe, the broadband dielectric broadband spectrometer, and a custom stripline resonator. Instant mashed potato is used as a food standard, which allowing for a large set of samples to be measured with various moisture contents. When directly compared at 1 GHz, all three methods produced comparable dielectric properties for 20% and 30% potato mixtures.

Compact Symmetric Objects (CSOs) form a distinct class of jetted active galactic nuclei (jetted-AGN). We examine a carefully selected sample of 54 CSOs, and confirm that there are two unrelated classes: an edge-dimmed, low-luminosity class (CSO 1), and an edge-brightened, high-luminosity class (CSO 2). Using statistically significant blind tests, we show that CSO 2s themselves consist of two morphologically distinct classes: CSO 2.0, having prominent hot-spots at the leading edges of narrow jets and/or narrow lobes; and CSO 2.2, without prominent hot-spots, and with broad jets and/or lobes. An intermediate class, CSO 2.1, exhibits mixed properties. The four classes occupy different, overlapping, portions of the luminosity-size plane, with the sizes of largest CSOs being $\sim 500$pc. We advance the hypothesis that CSO 2.0s are young and evolve through CSO 2.1s into CSO 2.2s, which are old (up to $\sim 5000$ yr). Thus CSOs do not evolve into larger types of jetted-AGN, but spend their whole life cycle as CSOs. The radio emission region energies in the CSO 2s we have studied range from $\sim 10^{-4}\, M_\odot {c}^2$ to $\sim 7 \, M_\odot {c}^2$. We show that the transient nature of CSO 2s, and their birthrate, can be explained through ignition in the tidal disruption events of giant stars. We also consider possibilities for tapping the spin energy of the supermassive black hole, and tapping the energy of the accretion disk, in a manner similar to, but not the same as, that which occurs in dwarf novae. Our results demonstrate conclusively that CSOs constitute a large family of AGN in which we have thus far studied only the brightest. More comprehensive radio studies, with higher sensitivity, resolution, and dynamic range, will revolutionize our understanding of AGN and the central engines that power them., Comment: 41 pages, 13 figures, 10 tables

We introduce a new Bravais lattice determination algorithm. SELLA is a straight-forward algorithm and a program for determining Bravais lattice type based on Selling (Delone) reduction. It is a complete, closed solution, and it provides a clear metric of fit to each type., Comment: 20 pages, 7 figures

The Delone (Selling) scalars, which are used in unit cell reduction and in lattice type determination, are studied in $\mathbb{C}^3$, the space of three complex variables. The three complex coordinate planes are composed of the six Delone scalars., Comment: 8 pages, 5 figures, revision added the 5 figures and section 5. Note processing added copy of a figure as decoration on 9th page

Unit cells are used to represent crystallographic lattices. Calculations measuring the differences between unit cells are used to provide metrics for measuring meaningful distances between three-dimensional crystallographic lattices. This is a surprisingly complex and computationally demanding problem. We present a review of the current best practice using Delaunay-reduced unit cells in the six-dimensional real space of Selling scalar cells S6 and the equivalent three-dimensional complex space C3. The process is a simplified version of the process needed when working with the more complex six-dimensional real space of Niggli-reduced unit cells G6. Obtaining a distance begins with identification of the fundamental region in the space, continues with conversion to primitive cells and reduction, analysis of distances to the boundaries of the fundamental unit, and is completed by a comparison of direct paths to boundary-interrupted paths, looking for a path of minimal length., Comment: 45 pages, 24 figures

For a real number $k$, define $\pi_k(x) = \sum_{p\le x} p^k$. When $k>0$, we prove that $$ \pi_k(x) - \pi(x^{k+1}) = \Omega_{\pm}\left(\frac{x^{\frac12+k}}{\log x} \log\log\log x\right) $$ as $x\to\infty$, and we prove a similar result when $-10$ and usually positive when $-1

This paper studies Iwasawa invariants in anti-cyclotomic towers. We do this by proposing two heuristics supported by computations. First we propose the Intersection Heuristics: these model `how often' the $p$-Hilbert class field of an imaginary quadratic field intersects the anti-cyclotomic tower and to what extent. Second we propose the Invariants Heuristics: these predict that the Iwasawa invariants $\lambda$ and $\mu$ usually vanish for imaginary quadratic fields where $p$ is non-split., Comment: v2: Incorporated changes as per the referee reports; accepted for publication (Experimental Math)

Characterization of crystallographic lattices is an important tool in structure solution, crystallographic database searches and clustering of diffraction images in serial crystallography. Characterization of lattices by Niggli-reduced cells (based on the three shortest non-coplanar lattice edge vectors) or by Delaunay-reduced cells (based on four edge vectors summing to zero and all meeting at obtuse or right angles) are commonly used. The Niggli cell derives from Minkowski reduction. The Delaunay cell derives from Selling reduction. All are related to the Wigner-Seitz (or Dirichlet, or Voronoi) cell of the lattice, which consists of the points at least as close to a chosen lattice point than they are to any other lattice point. Starting from a Niggli-reduced cell, the Dirichlet cell is characterized by the planes determined by thirteen lattice half-edges: the midpoints of the three Niggli cell edges, the six Niggli cell face diagonals and the four body-diagonals, but seven of the edge lengths are sufficient: three edge lengths, the three shorter of each pair of face-diagonal lengths and the shortest body-diagonal length, from which the Niggli-reduced cell may be recovered., Comment: 37 pages, 2 figures, general copy-edit, update matrices to full 2D display

We have formalised Szemer\'edi's Regularity Lemma and Roth's Theorem on Arithmetic Progressions, two major results in extremal graph theory and additive combinatorics, using the proof assistant Isabelle/HOL. For the latter formalisation, we used the former to first show the Triangle Counting Lemma and the Triangle Removal Lemma: themselves important technical results. Here, in addition to showcasing the main formalised statements and definitions, we focus on sensitive points in the proofs, describing how we overcame the difficulties that we encountered.

The formalisation of mathematics is continuing rapidly, however combinatorics continues to present challenges to formalisation efforts, such as its reliance on techniques from a wide range of other fields in mathematics. This paper presents formal linear algebraic techniques for proofs on incidence structures in Isabelle/HOL, and their application to the first formalisation of Fisher's inequality. In addition to formalising incidence matrices and simple techniques for reasoning on linear algebraic representations, the formalisation focuses on the linear algebra bound and rank arguments. These techniques can easily be adapted for future formalisations in combinatorics, as we demonstrate through further application to proofs of variations on Fisher's inequality., Comment: Accepted to ITP 2022, to be published in conference proceedings

We adapt a known technique for searching for ideal classes of arbitrary order and then apply it to three families of number fields. We show that a family of cyclic sextic number fields has infinitely many fields in it that contain a relative ideal class of order $r,$ where $r$ is a positive integer relatively prime to the degree of the extension. We then show that the same holds true for a family of cyclic quartic number fields. Though the technique is traditionally applied to Galois extensions, we show how it may be adapted to handle a family of non-Galois cubic number fields and prove that this family contains infinitely many fields with an ideal class of arbitrary order relatively prime to three., Comment: 23 pages