ShantanuT01/DACTYL-Pretraining huggingface dataset:
Using a generalization due to Lerch [M. Lerch, Sur un th\'{e}or\`{e}me de Zolotarev. Bull. Intern. de l'Acad. Fran\c{c}ois Joseph 3 (1896), 34-37] of a classical lemma of Zolotarev, employed in Zolotarev's proof of the law of quadratic reciprocity, we determine necessary and sufficient conditions for the difference of two Dedekind sums to be in $8\mathbb{Z}$. These yield new necessary conditions for equality of two Dedekind sums. In addition, we resolve a conjecture of Girstmair [Girstmair, Congruences mod 4 for the alternating sum of the partial quotients, arXiv: 1501.00655].
Measurement interaction between a measured object and a measuring instrument, if both are initially in a pure state, produces a (final) bipartite entangled state vector. The quasi-classical part of the correlations in it is connected with transmission of information in the measurement. But, prior to "reading" the instrument, there is also purely quantum entanglement in the final state vector. It is shown that in repeatable measurement quantitatively the entanglement equals the amount of incompatibility between the measured observable and the final state. It also equals the amount of incompatibility of the observable and the initial state of the object.
We consider the effect of a periodic perturbation with frequency $\omega$ on the holographic N=4 plasma represented by the planar AdS black hole. The response of the system is given by exponentially decaying waves. The corresponding complex wave numbers can be found by solving wave equations in the AdS black hole background with infalling boundary conditions on the horizon in an analogous way as in the calculation of quasinormal modes. The complex momentum eigenvalues have an interpretation as poles of the retarded Green's functions, where the inverse of the imaginary part gives an absorption length $\lambda$. At zero frequency we obtain the screening length for a static field. These are directly related to the glueball masses in the dimensionally reduced theory. We also point out that the longest screening length corresponds to an operator with non-vanishing R-charge and thus does not have an interpretation as a QCD3 glueball.
We confirm the equitable $\Delta$-coloring conjecture for interval graphs and establish the monotonicity of equitable colorability for them. We further obtain results on equitable colorability about square (or Cartesian) and cross (or direct) products of graphs.
We investigated optical spectra of quasi-two-dimensional multi-band Ca$% _{2-x} $Sr$_{x}$RuO$_{4}$ systems. The extended Drude model analysis on the ab-plane optical conductivity spectra indicates that the effective mass should be enhanced near $x=0.5$. Based on the sum rule argument, we showed that the orbital-selective Mott-gap opening for the $d_{yz/zx}$ bands, the widely investigated picture, could not be the origin of the mass enhancement. We exploited the multi-band effects in the extended Drude model analysis, and demonstrated that the intriguing heavy mass state near $x=0.5$ should come from the renormalization of the $d_{xy}$ band.
The Hyades, Praesepe, and Pleiades are well studied stellar clusters that anchor important secondary stellar age indicators. Recent studies have shown that main sequence turn off-based ages for these clusters may depend on the degree of rotation in the underlying stellar models. Rotation induces structural instabilities that can enhance the chemical mixing of a star, extending its fuel supply. In addition, rotation introduces a modulation of the star's observed magnitude and color due to the effects of gravity darkening. We aim to investigate the extent to which stellar rotation affects the age determination of star clusters. We utilize the MESA stellar evolution code to create models that cover a range of rotation rates corresponding to $\Omega/\Omega_c=0.0$ to $0.6$ in $0.1$ dex steps, allowing the assessment of variations in this dimension. The statistical analysis package, MATCH, is employed to derive ages and metallicities by fitting our MESA models to Tycho $B_T$, $V_T$ and 2MASS $J$, $K_s$ color-magnitude diagrams. We find that the derived ages are relatively insensitive to the effects of rotation. For the Hyades, Praesepe, and Pleiades, we derive ages based on synthetic populations that model a distribution of rotation rates or a fixed rate. Across each case, derived ages tend to agree roughly within errors, near $680$, $590$, and $110-160$ Myr for the Hyades, Praesepe, and Pleiades, respectively. These ages are in agreement with Li depletion boundary-based ages and previous analyses that used non-rotating isochrones. Our methods do not provide a strong constraint on the metallicities of these clusters.
Finite-temperature micromagnetics simulations are employed to study the magnetization-switching dynamics driven by a field applied at an angle to the long axis of an iron nanopillar. A bi-modal distribution in the switching times is observed, and evidence for two competing modes of magnetization-switching dynamics is presented. For the conditions studied here, temperature $T = 20$ K and the reversal field 3160 Oe at an angle of 75$^\circ$ to the long axis, approximately 70% of the switches involve unstable decay (no free-energy barrier) and 30% involve metastable decay (a free-energy barrier is crossed). The latter are indistinguishable from switches which are constrained to start at a metastable free-energy minimum. Competition between unstable and metastable decay could greatly complicate applications involving magnetization switches near the coercive field.
The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as sums over all spanning subgraphs, as sums over spanning subgraphs with no broken circuits, and in terms of acyclic orientations with compatible colorings. We establish all six of these expressions bijectively. In fact, we do this with only two bijections, as the proofs in the symmetric function setting are obtained using the same bijections as in the polynomial case and the bijection for broken circuits is just a restriction of the one for all spanning subgraphs.
Magnetic fields in cool stars can be investigated by measuring Zeeman line broadening and polarization in atomic and molecular lines. Similar to the Sun, these fields are complex and height-dependent. Many molecular lines dominating M-dwarf spectra (e.g., FeH, CaH, MgH, and TiO) are temperature -- and Zeeman -- sensitive and form at different atmospheric heights, which makes them excellent probes of magnetic fields on M dwarfs. Our goal is to analyze the complexity of magnetic fields in M dwarfs. We investigate how magnetic fields vary with the stellar temperature and how "surface" inhomogeneities are distributed in height -- the dimension that is usually neglected in stellar magnetic studies. We have determined effective temperatures of the photosphere and of magnetic features, magnetic field strengths and filling factors for nine M dwarfs (M1-M7). Our chi^2 analysis is based on a comparison of observed and synthetic intensity and circular polarization profiles. Stokes profiles were calculated by solving polarized radiative transfer equations. Properties of magnetic structures depend on the analyzed atomic or molecular species and their formation heights. Two types of magnetic features similar to those on the Sun have been found: a cooler (starspots) and a hotter (network) one. The magnetic field strength in both starspots and network is within 3 kG to 6 kG, on average it is 5 kG. These fields occupy a large fraction of M dwarf atmospheres at all heights, up to 100%. The plasma beta is less than one, implying highly magnetized stars. A combination of molecular and atomic species and a simultaneous analysis of intensity and circular polarization spectra have allowed us to better decipher the complexity of magnetic fields on M dwarfs, including their dependence on the atmospheric height. This work provides an opportunity to investigate a larger sample of M dwarfs and L-type brown dwarfs.
We investigate the dependence of the peak of the IMF on the physics of the so-called first Larson core, which corresponds to the point where the dust becomes opaque to its own radiation. We perform numerical simulations of collapsing clouds of $1000 M_\odot$ for various gas equation of state (eos), paying great attention to the numerical resolution and convergence. The initial conditions of these numerical experiments are varied in the companion paper. We also develop analytical models that we confront to our numerical results. If an isothermal eos is used, we show that the peak of the IMF shifts to lower masses with improved numerical resolution. When an adiabatic eos is employed, numerical convergence is obtained. The peak position varies with the eos and we find that the peak position is about ten times the mass of the first Larson core. By analyzing the stability of non-linear density fluctuations in the vicinity of a point mass and then summing over a reasonable density distribution, we find that tidal forces exert a strong stabilizing effect and likely lead to a preferential mass several times larger than that of the first Larson core. We propose that in a sufficiently massive and cold cloud, the peak of the IMF is determined by the thermodynamics of the high density adiabatic gas as well as the stabilizing influence of tidal forces. The resulting characteristic mass is about ten times the mass of the first Larson core, which altogether leads to a few tenths of solar masses. Since these processes are not related to the large scale physical conditions and to the environment, our results suggest a possible explanation for the apparent universality of the peak of the IMF.
Stars in open clusters are expected to share an identical abundance pattern. Establishing the level of chemical homogeneity in a given open cluster deserves further study as it is the basis of the concept of chemical tagging to unravel the history of the Milky Way. M67 is particularly interesting given its solar metallicity and age as well as being a dense cluster environment. We conducted a strictly line-by-line differential chemical abundance analysis of two solar twins in M67: M67-1194 and M67-1315. Stellar atmospheric parameters and elemental abundances were obtained with high precision using Keck/HIRES spectra. M67-1194 is essentially identical to the Sun in terms of its stellar parameters. M67-1315 is warmer than M67-1194 by ~ 150 K as well as slightly more metal-poor than M67-1194 by ~ 0.05 dex. M67-1194 is also found to have identical chemical composition to the Sun, confirming its solar twin nature. The abundance ratios [X/Fe] of M67-1315 are similar to the solar abundances for elements with atomic number Z <= 30, while most neutron-capture elements are enriched by ~ 0.05 dex, which might be attributed to enrichment from a mixture of AGB ejecta and r-process material. The distinct chemical abundances for the neutron-capture elements in M67-1315 and the lower metallicity of this star compared to M67-1194, indicate that the stars in M67 are likely not chemically homogeneous. This poses a challenge for the concept of chemical tagging since it is based on the assumption of stars forming in the same star-forming aggregate.
The ground state of a two-dimensional (2D) system of Bose particles of spin zero, interacting via a repulsive Gaussian-Core potential, has been investigated by means of Quantum Monte Carlo simulations. The quantum phase diagram is qualitatively identical to that of 2D Yukawa bosons. While the system is a fluid at all densities for weak coupling, in the strong coupling regime it transitions upon compression from a low density superfluid to a crystal, and then into a reentrant superfluid phase. No evidence of a (supersolid) cluster crystal phase is seen.
Strange baryons have long been known to exhibit a leading particle effect. A recent comparison of $\Xi^-$ production in $\pi^-$, $n$, and $\Sigma^-$ interactions with nuclei show this effect clearly. These data are supplemented by earlier measurements of $\Xi^-$ and $\Omega$ production by a $\Xi^-$ beam. We calculate the $\Xi^-$ and $\Omega$ $x_F$ distributions and nuclear dependence in $hA$ interactions using the intrinsic model.
In this paper, we are interested in the nonlinear Rayleigh-Taylor instability for the gravity-driven incompressible Navier-Stokes equations with Navier-slip boundary conditions around a smooth increasing density profile $\rho_0(x_2)$ in a slab domain $2\pi L\mathbb{T} \times (-1,1)$ ($L>0$, $\mathbb{T}$ is the usual 1D torus). The linear instability study of the viscous Rayleigh-Taylor model amounts to the study of the following ODE on the finite interval $(-1,1)$, \begin{equation}\label{EqMain} \lambda^2 ( \rho_0 k^2 \phi - (\rho_0 \phi')')+ \lambda \mu (\phi^{(4)} - 2k^2 \phi'' + k^4 \phi) = gk^2 \rho_0'\phi, \end{equation} with the boundary conditions \begin{equation}\label{4thBound} \begin{cases} \phi(-1)=\phi(1)=0,\\ \mu \phi''(1) = \xi_+ \phi'(1), \\ \mu \phi''(-1) =- \xi_- \phi'(-1), \end{cases} \end{equation} where $\lambda>0$ is the growth rate in time, $g>0$ is the gravity constant, $k$ is the wave number and two Navier-slip coefficients $\xi_{\pm}$ are nonnegative constants. For each $k\in L^{-1}\mathbb{Z}$, we define a threshold of viscosity coefficient $\mu_c(k,\Xi)$ for the linear instability. So that, in the $k$-supercritical regime, i.e. $\mu>\mu_c(k,\Xi)$, we describe a spectral analysis adapting the operator method initiated by Lafitte-Nguyen \cite{LN20} and prove that there are infinite nontrivial solutions $(\lambda_n, \phi_n)_{n\geqslant 1} $ of \eqref{EqMain}-\eqref{4thBound} with $\lambda_n \to 0$ as $n\to \infty$ and $\phi_n\in H^4((-1,1))$. Based on the existence of infinitely many normal modes of the linearized problem, we construct a wide class of initial data to the nonlinear equations, extending the previous framework of Guo-Strauss \cite{GS95} and of Grenier \cite{Gre00}, to prove the nonlinear Rayleigh-Taylor instability in a high regime of viscosity coefficient, namely $\mu >3\sup_{k\in L^{-1}\mathbb{Z}\setminus\{0\}}\mu_c(k,\Xi)$.
We investigate lamellar three-phase patterns that form during the directional solidification of ternary eutectic alloys in thin samples. A distinctive feature of this system is that many different geometric arrangements of the three phases are possible, contrary to the widely studied two-phase patterns in binary eutectics. Here, we first analyze the case of stable lamellar coupled growth of a symmetric model ternary eutectic alloy, using a Jackson-Hunt type calculation in thin film morphology, for arbitrary configurations, and derive expressions for the front undercooling as a function of velocity and spacing. Next, we carry out phase-field simulations to test our analytic predictions and to study the instabilities of the simplest periodic lamellar arrays. For large spacings, we observe different oscillatory modes that are similar to those found previously for binary eutectics and that can be classified using the symmetry elements of the steady-state pattern. For small spacings, we observe a new instability that leads to a change in the sequence of the phases. Its onset can be well predicted by our analytic calculations. Finally, some preliminary phase-field simulations of three-dimensional growth structures are also presented.
The EUV (100-912 {\AA}) is a spectral region notoriously difficult to observe due to attenuation by neutral hydrogen gas in the interstellar medium. Despite this, hundreds to thousands of nearby stars of different spectral types and magnetic activity levels are accessible in the EUV range. The EUV probes interesting and complicated regions in the stellar atmosphere like the lower corona and transition region that are inaccessible from other spectral regions. In this white paper we describe how direct EUV observations, which require a dedicated grazing-incidence observatory, cannot yet be accurately substituted with models and theory. Exploring EUV emission from cool dwarf stars in the time domain can make a major contribution to understanding stellar outer atmospheres and magnetism, and offers the clearest path toward detecting coronal mass ejections on stars other than the Sun.
Observational evidence suggests that the majority of stars may have been born in stellar clusters or associations. Within these dense environments, dynamical interactions lead to high rates of close stellar encounters. A variety of recent observational and theoretical indications suggest stellar-mass black holes may be present and play an active dynamical role in stellar clusters of all masses. In this study, we explore the tidal disruption of main sequence stars by stellar-mass black holes in young star clusters. We compute a suite of over 3000 independent $N$-body simulations that cover a range in cluster mass, metallicity, and half-mass radii. We find stellar-mass black hole tidal disruption events (TDEs) occur at an overall rate of up to roughly $200\,\rm{Gpc}^{-3}\,\rm{yr}^{-1}$ in young stellar clusters in the local universe. These TDEs are expected to have several characteristic features, namely fast rise times of order a day, peak X-ray luminosities of at least $10^{44}\,\rm{erg\,s}^{-1}$, and bright optical luminosities (roughly $10^{41}-10^{44}\,\rm{erg\,s}^{-1}$) associated with reprocessing by a disk wind. In particular, we show these events share many features in common with the emerging class of Fast Blue Optical Transients.
The historic first joint detection of both gravitational wave and electromagnetic emission from a binary neutron star merger cemented the association between short gamma-ray bursts (SGRBs) and compact object mergers, as well as providing a well sampled multi-wavelength light curve of a radioactive kilonova (KN) for the first time. Here we compare the optical and near-infrared light curves of this KN, AT2017gfo, to the counterparts of a sample of nearby (z < 0.5) SGRBs to characterize their diversity in terms of their brightness distribution. Although at similar epochs AT2017gfo appears fainter than every SGRB-associated KN claimed so far, we find three bursts (GRBs 050509B, 061201 and 080905A) where, if the reported redshifts are correct, deep upper limits rule out the presence of a KN similar to AT2017gfo by several magnitudes. Combined with the properties of previously claimed KNe in SGRBs this suggests considerable diversity in the properties of KN drawn from compact object mergers, despite the similar physical conditions that are expected in many NS-NS mergers. We find that observer angle alone is not able to explain this diversity, which is likely a product of the merger type (NS-NS versus NS-BH) and the detailed properties of the binary (mass ratio, spins etc). Ultimately disentangling these properties should be possible through observations of SGRBs and gravitational wave sources, providing direct measurements of heavy element enrichment throughout the Universe.
This paper presents a shape-theoretic framework for dynamical analysis of nonlinear dynamical systems which appear frequently in several video-based inference tasks. Traditional approaches to dynamical modeling have included linear and nonlinear methods with their respective drawbacks. A novel approach we propose is the use of descriptors of the shape of the dynamical attractor as a feature representation of nature of dynamics. The proposed framework has two main advantages over traditional approaches: a) representation of the dynamical system is derived directly from the observational data, without any inherent assumptions, and b) the proposed features show stability under different time-series lengths where traditional dynamical invariants fail. We illustrate our idea using nonlinear dynamical models such as Lorenz and Rossler systems, where our feature representations (shape distribution) support our hypothesis that the local shape of the reconstructed phase space can be used as a discriminative feature. Our experimental analyses on these models also indicate that the proposed framework show stability for different time-series lengths, which is useful when the available number of samples are small/variable. The specific applications of interest in this paper are: 1) activity recognition using motion capture and RGBD sensors, 2) activity quality assessment for applications in stroke rehabilitation, and 3) dynamical scene classification. We provide experimental validation through action and gesture recognition experiments on motion capture and Kinect datasets. In all these scenarios, we show experimental evidence of the favorable properties of the proposed representation.
The final chapter in the long-standing mystery of the gamma-ray bursts (GRBs) centres on the origin of the short-hard class, suspected on theoretical grounds to result from the coalescence of neutron star or black hole binary systems. Numerous searches for the afterglows of short-hard bursts have been made, galvanized by the revolution in our understanding of long-duration GRBs that followed the discovery in 1997 of their broadband (X-ray, optical, and radio) afterglow emission. Here we present the discovery of the X-ray afterglow of a short-hard burst whose accurate position allows us to unambiguously associate it with a star-forming galaxy at redshift z=0.160, and whose optical lightcurve definitively excludes a supernova association. Together with results from three other recent short-hard bursts, this suggests that short-hard bursts release much less energy than the long-duration GRBs. Models requiring young stellar populations, such as magnetars and collapsars, are ruled out, while coalescing degenerate binaries remain the most promising progenitor candidates.
We introduce an ordinal classification algorithm for photometric redshift estimation, which significantly improves the reconstruction of photometric redshift probability density functions (PDFs) for individual galaxies and galaxy samples. As a use case we apply our method to CFHTLS galaxies. The ordinal classification algorithm treats distinct redshift bins as ordered values, which improves the quality of photometric redshift PDFs, compared with non-ordinal classification architectures. We also propose a new single value point estimate of the galaxy redshift, that can be used to estimate the full redshift PDF of a galaxy sample. This method is competitive in terms of accuracy with contemporary algorithms, which stack the full redshift PDFs of all galaxies in the sample, but requires orders of magnitudes less storage space.   The methods described in this paper greatly improve the log-likelihood of individual object redshift PDFs, when compared with a popular Neural Network code (ANNz). In our use case, this improvement reaches 50\% for high redshift objects ($z \geq 0.75$).   We show that using these more accurate photometric redshift PDFs will lead to a reduction in the systematic biases by up to a factor of four, when compared with less accurate PDFs obtained from commonly used methods. The cosmological analyses we examine and find improvement upon are the following: gravitational lensing cluster mass estimates, modelling of angular correlation functions, and modelling of cosmic shear correlation functions.
Given a unimodular lattice $\Lambda\subseteq \mathbb{R}^2$ consider the counting function $\mathcal{N}_\Lambda(T)$ counting the number of lattice points of norm less than $T$, and the remainder $\mathcal{R}_\Lambda(T)=\mathcal{N}(T)-\pi T^2$. We give an elementary proof that the mean square of the remainder over the set of all shears of a unimodular lattice is bounded by $O(T\log^2(T))$.