ShantanuT01/DACTYL-Pretraining huggingface dataset:
We develop a numerical code to compute gravitational waves induced by a particle moving on eccentric inclined orbits around a Kerr black hole. For such systems, the black hole perturbation method is applicable. The gravitational waves can be evaluated by solving the Teukolsky equation with a point like source term, which is computed from the stress-energy tensor of a test particle moving on generic bound geodesic orbits. In our previous papers, we computed the homogeneous solutions of the Teukolsky equation using a formalism developed by Mano, Suzuki and Takasugi and showed that we could compute gravitational waves efficiently and very accurately in the case of circular orbits on the equatorial plane. Here, we apply this method to eccentric inclined orbits. The geodesics around a Kerr black hole have three constants of motion: energy, angular momentum and the Carter constant. We compute the rates of change of the Carter constant as well as those of energy and angular momentum. This is the first time that the rate of change of the Carter constant has been evaluated accurately. We also treat the case of highly eccentric orbits with $e=0.9$. To confirm the accuracy of our codes, several tests are performed. We find that the accuracy is only limited by the truncation of $\ell$-, $k$- and $n$-modes, where $\ell$ is the index of the spin-weighted spheroidal harmonics, and $n$ and $k$ are the harmonics of the radial and polar motion, respectively. When we set the maximum of $\ell$ to 20, we obtain a relative accuracy of $10^{-5}$ even in the highly eccentric case of $e=0.9$. The accuracy is better for lower eccentricity. Our numerical code is expected to be useful for computing templates of the extreme mass ratio inspirals, which is one of the main targets of the Laser Interferometer Space Antenna (LISA).
We consider what is the best way to extract science from large surveys of the Milky Way galaxy. The diversity of data gathered in these surveys, together with our position within the Galaxy, imply that science must be extracted by fitting dynamical models to the data in the space of the observables. Models based on orbital tori promise to be superior for this task than traditional types of models, such as N-body models and Schwarzschild models. A formalism that allows such models to be fitted to data is developed and tested on pseudodata of varying richness.
Diophantine triples taking values in recurrence sequences have recently been studied quite a lot. In particular the question was raised whether or not there are finitely many Diophantine triples in the Tribonacci sequence. We answer this question here in the affirmative. We prove that there are only finitely many triples of integers $1\le u<v<w$ such that $uv+1,uw+1,vw+1$ are Tribonacci numbers. The proof depends on the Subspace theorem.
We develop a theory for fluctuations and correlations in a gas evolving under ballistic annihilation dynamics. Starting from the hierarchy of equations governing the evolution of microscopic densities in phase space, we subsequently restrict to a regime of spatial homogeneity, and obtain explicit predictions for the fluctuations and time correlation of the total number of particles, total linear momentum and total kinetic energy. Cross-correlations between these quantities are worked out as well. These predictions are successfully tested against Molecular Dynamics and Monte-Carlo simulations. This provides strong support for the theoretical approach developed, including the hydrodynamic treatment of the spectrum of the linearized Boltzmann operator. This article is a companion paper to arXiv:0801.2299 and makes use of the spectral analysis reported there.
We present a reconstruction theorem for Fano vector bundles on projective space which recovers the small quantum cohomology for the projectivisation of the bundle from a small number of low-degree Gromov--Witten invariants. We provide an extended example in which we calculate the quantum cohomology of a certain Fano 9-fold and deduce from this, using the quantum Lefschetz theorem, the quantum period sequence for a Fano 3-fold of Picard rank 2 and degree 24. This example is new, and is important for the Fanosearch programme.
In this paper, we extend the gravitational bending of light studies in Kottler metrics to comprise nonlinear electrodynamics within the framework of Einstein - power - Maxwell theory. We show that the closest approach distance and the gravitational bending of light are affected from the presence of charge for particular values of the power parameter $k$, which is defined by means of energy conditions. It is shown that the bending angle of light is stronger in the case of a strong electric field, which is the case for $k=1.2$.
This paper presents 828 new 21 cm neutral hydrogen line measurements carried out with the FORT receiver of the meridian transit Nan\c{c}ay radio telescope (NRT) in the years 2000 -- 2007. This observational program was part of a larger project aimed at collecting an exhaustive and magnitude-complete HI extragalactic catalog for Tully-Fisher applications. Through five massive data releases, the KLUN series has collected a homogeneous sample of 4876 HI-spectra of spiral galaxies, complete down to a flux of 5 Jy.km.s^{-1} and with declination delta > -40{\deg}. We publish here the last release of the KLUN HI observational program, corresponding to the faint end of the survey, with HI masses ranging from 5 10^8 to 5 10^{10} solar masses. The size of this final sample is comparable to the catalogs based on the Arecibo and Parkes radio telescope campaigns, and it allows general HI mass distribution studies from a set of homogeneous radio measurements.
We show that K\"uchle fivefolds of type (c5) --- subvarieties of the Grassmannian Gr(3,7) parameterizing 3-subspaces that are isotropic for a given 2-form and are annihilated by a given 4-form --- are birational to hyperplane sections of the Lagrangian Grassmannian LGr(3,6) and describe in detail these birational transformations. As an application, we show that the integral Chow motive of a K\"uchle fivefold of type (c5) is of Lefschetz type. We also discuss K\"uchle fourfolds of type (c5) --- hyperplane sections of the corresponding K\"uchle fivefolds --- an interesting class of Fano fourfolds, which is expected to be similar to the class of cubic fourfolds in many aspects.
We study the local regularity of $p$-caloric functions or more generally of $\phi$-caloric functions. In particular, we study local solutions of non-linear parabolic systems with homogeneous right hand side, where the leading terms has Uhlenbeck structure of Orlicz type. This paper closes the gap of [22] where Liebermann proved that if the gradient of a solution is bounded, it is H\"older continuous.   The crucial step is a novel local estimates for the gradient of the solutions, which generalize and improve the pioneering estimates of DiBenedetto and Friedman [12,10] for the $p$-Laplace heat equation.
In this paper, we have studied the Klein's paradox in the presence of both scalar and vector potential barriers. From the corresponding Dirac equation we have calculated the transmission and reflection coefficients. It is shown that the presence of a scalar barrier the scalar potential wides the gap between positive and negative energies and so the forbidden region. Accordingly, the Klein's paradox disappears when the scalar barrier exceeds a critical value. Considering the problem within the framework of quantum field theory, we have calculated the related pair creation probability, the mean number of created particles and the probability of a vacuum to remain a vacuum. Then it is shown that the scalar potential cut down the Klein range and minimizes the creation of particles; The particle creation decreases as the scalar potential increases and ceases definitely when the scalar potential reaches the critical value.
In the present paper, we discuss solvability questions of a non-local problem with integral form transmitting conditions for diffusion-wave equation with the Caputo fractional derivative in a domain bounded by smooth curves. The uniqueness of the solution of the formulated problem we prove using energy integral method with some modifications. The existence of solution will be proved by equivalent reduction of the studied problem into a system of second kind Fredholm integral equations.
We model the diffusive shock acceleration of particles in a system of two colliding shock waves and present a method to solve the time-dependent problem analytically in the test-particle approximation and high energy limit. In particular, we show that in this limit the problem can be analysed with the help of a self-similar solution. While a number of recent works predict hard ($E^{-1}$) spectra for the accelerated particles in the stationary limit, or the appearance of spectral breaks, we found instead that the spectrum of accelerated particles in a time-dependent collision follows quite closely the canonical $E^{-2}$ prediction of diffusive shock acceleration at a single shock, except at the highest energy, where a hardening appears, originating a bumpy feature just before the exponential cutoff. We also investigated the effect of the reacceleration of pre-existing cosmic rays by a system of two shocks, and found that under certain conditions spectral features can appear in the cutoff region. Finally, the mathematical methods presented here are very general and could be easily applied to a variety of astrophysical situations, including for instance standing shocks in accretion flows, diverging shocks, backward collisions of a slow shock by a faster shock, and wind-wind or shock-wind collisions.
Bicliques are complements of bipartite graphs; as such each consists of two cliques joined by a number of edges. In this paper we study algebraic aspects of the chromatic polynomials of these graphs. We derive a formula for the chromatic polynomial of an arbitrary biclique, and use this to give certain conditions under which two of the graphs have chromatic polynomials with the same splitting field. Finally, we use a subfamily of bicliques to prove the cubic case of the $\alpha +n$ conjecture, by showing that for any cubic integer $\alpha$, there is a natural number $n$ such that $\alpha +n$ is a chromatic root.
Fractional diffusion equations for three-dimensional lattice models based on fractional-order differences of the Grunwald-Letnikov type are suggested. These lattice fractional diffusion equations contain difference operators that describe long-range jumps from one lattice site to other. In continuum limit, the suggested lattice diffusion equations with non-integer order differences give the diffusion equations with the Grunwald-Letnikov fractional derivatives for continuum. We propose a consistent derivation of the fractional diffusion equation with the fractional derivatives of Grunwald-Letnikov type. The suggested lattice diffusion equations can be considered as a new microstructural basis of space-fractional diffusion in nonlocal media.
Parkes pulse arrival-time data for 165 radio pulsars spanning from 1990 to 2011 have been searched for period glitches. Forty-six events out of the detected 107 glitches were found to be new contributions to the entire glitch population which currently contains approximately 400 events.
In the regime of weakly transverse long waves, given long-wave initial data, we prove that the nondimensionalized water wave system in an infinite strip under influence of gravity and surface tension on the upper free interface has a unique solution on $[0,{T}/\eps]$ for some $\eps$ independent of constant $T.$ We shall prove in the subsequent paper \cite{MZZ2} that on the same time interval, these solutions can be accurately approximated by sums of solutions of two decoupled Kadomtsev-Petviashvili (KP) equations.
The tree-depth of $G$ is the smallest value of $k$ for which a labeling of the vertices of $G$ with elements from $\{1,\dots,k\}$ exists such that any path joining two vertices with the same label contains a vertex having a higher label. The graph $G$ is $k$-critical if it has tree-depth $k$ and every proper minor of $G$ has smaller tree-depth.   Motivated by a conjecture on the maximum degree of $k$-critical graphs, we consider the property of 1-uniqueness, wherein any vertex of a critical graph can be the unique vertex receiving label 1 in an optimal labeling. Contrary to an earlier conjecture, we construct examples of critical graphs that are not 1-unique and show that 1-unique graphs can have arbitrarily many more edges than certain critical spanning subgraphs. We also show that $(n-1)$-critical graphs are 1-unique and use 1-uniqueness to show that the Andr\'{a}sfai graphs are critical with respect to tree-depth.
We study the dynamics of density fluctuations in the steady state of a non-equilibrium system, the Zero-Range Process on a ring lattice. Measuring the time series of the total number of particles in a \emph{segment} of the lattice, we find remarkable structures in the associated power spectra, namely, two distinct components of damped-oscillations. The essential origin of both components is shown in a simple pedagogical model. Using a more sophisticated theory, with an effective drift-diffusion equation governing the stochastic evolution of the local particle density, we provide reasonably good fits to the simulation results. The effects of altering various parameters are explored in detail. Avenues for improving this theory and deeper understanding of the role of particle interactions are indicated.
We report on the detection with the Infrared Space Observatory (ISO) of a molecular band at 57.5 microns (174 cm^{-1}) in carbon-rich evolved stars and in Sgr B2. Taking into account the chemistry of these objects the most likelihood carrier is a carbon chain. We tentatively assign the band to the nu_5 bending mode of C_4 for which a wavenumber of 170-172.4 cm^{-1} has been derived in matrix experiments (Withey et al. 1991). An alternate carrier might be C_4H, although the frequency of its lowest energy vibrational bending mode, nu_7, is poorly known (130-226 cm^{-1}). If the carrier is C_4, the derived maximum abundance is nearly similar to that found for C_3 in the interstellar and circumstellar media by Cernicharo, Goicoechea & Caux (2000). Hence, tetra-atomic carbon could be one of the most abundant carbon chain molecules in these media.
It is shown that the transformation between ordinary and noncommutative Yang-Mills theory as formulated by Seiberg and Witten is due to the equivalence of certain star products on the D-brane world-volume.
Spin-orbit mediated phonon relaxation in a two-dimensional quantum dot is investigated using different confining potentials. Elliptical harmonic oscillator and cylindrical well results are compared to each other in the case of a two-electron GaAs quantum dot subjected to a tilted magnetic field. The lowest energy set of two-body singlet and triplet states are calculated including spin-orbit and magnetic effects. These are used to calculate the phonon induced transition rate from the excited triplet to the ground state singlet for magnetic fields up to where the states cross. The roll of the cubic Dresselhaus effect and the positioning of "spin hot-spots" are discussed and relaxation rates for a few different systems are exhibited.
We discuss some aspects related to the so-called Hilbert space Average Method, as an alternative to describe the dynamics of open quantum systems. First we present a derivation of the method which does not make use of the algebra satisfied by the operators involved in the dynamics, and extend the method to systems subject to a Hamiltonian that changes with time. Next we examine the performance of the adiabatic quantum search algorithm with a particular model for the environment. We relate our results to the criteria discussed in the literature for the validity of the above-mentioned method for similar environments.
A swift chemical route to synthesize Co-doped SnO$_2$ nanopowders is described. Pure and highly stable Sn$_{1-x}$Co$_x$O$_{2-\delta}$ (0 $\le$ x $\le$ 0.15) crystalline nanoparticles were synthesized, with mean grain sizes < 5 nm and the dopant element homogeneously distributed in substitutional sites of the SnO$_2$ matrix. The UV-visible diffuse reflectance spectra of the Sn$_{1-x}$Co$_x$O$_{2-\delta}$ samples reveal red shifts, the optical bandgap energies decreasing with increasing Co concentration. The Urbach energies of the samples were calculated and correlated with their bandgap energies. The photocatalytic activity of the Sn$_{1-x}$Co$_x$O$_{2-\delta}$ samples was investigated for the 4-hydroxylbenzoic acid (4-HBA) degradation process. A complete photodegradation of a 10 ppm 4-HBA solution was achieved using 0.02% (w/w) of Sn$_{0.95}$Co$_{0.05}$O$_{2-\delta}$ nanoparticles in 60 min of irradiation.
We set up recursion relations for the partition function and the ground-state occupancy for a fixed number of non-interacting bosons confined in a square box potential and determine the temperature dependence of the specific heat and the particle number in the ground state. A proper semiclassical treatment is set up which yields the correct small-T-behavior in contrast to an earlier theory in Feynman's textbook on Statistical Mechanics, in which the special role of the ground state was ignored. The results are compared with an exact quantum mechanical treatment. Furthermore, we derive the finite-size effect of the system.
We study numerically the effects of the Rashba spin-orbit interaction on the model of electrons in n-doped semiconductors of Matsubara and Toyozawa (MT). We focus on the analysis of the density of states (DOS) and the inverse participation ratio (IPR) of the spin-orbit perturbed states in the MT set of energy eigenstates in order to characterize the eigenstates with respect to their extended or localized nature. The finite sizes that we are able to consider necessitate an enhancement of the spin-orbit coupling strength in order to obtain a meaningful perturbation. The IPR and DOS are then studied as a function of the enhancement parameter.
In this paper, we consider a free boundary problem modeling the growth of spherically symmetric necrotic tumors with angiogenesis and a $\omega$-periodic supply $\phi(t)$ of external nutrients. In the model, the consumption rate of the nutrient and the proliferation rate of tumor cells $S(\sigma)$ are both general nonlinear functions. The well-posedness and asymptotic behavior of solutions are studied. We show that if the average of $S(\phi(t))$ is nonpositive, then all evolutionary tumors will finally vanish; the converse is also ture. If instead the average of $S(\phi(t))$ is positive, then there exists a unique positive periodic solution and all other evolutionary tumors will converge to this periodic state.
Nowadays, several opacity codes are able to provide data for stellar structure models, but the computed opacities may show significant differences. In this work, we present state-of-the-art precise spectral opacity calculations, illustrated by stellar applications. The essential role of laboratory experiments to check the quality of the computed data is underlined. We review some X-ray and XUV laser and Z-pinch photo-absorption measurements as well as X-ray emission spectroscopy experiments involving hot dense plasmas produced by ultra-high-intensity laser irradiation. The measured spectra are systematically compared with the fine-structure opacity code SCO-RCG. Focus is put on iron, due to its crucial role in understanding asteroseismic observations of $\beta$ Cephei-type and Slowly Pulsating B stars, as well as of the Sun. For instance, in $\beta$ Cephei-type stars, the iron-group opacity peak excites acoustic modes through the "kappa-mechanism". A particular attention is paid to the higher-than-predicted iron opacity measured at the Sandia Z-machine at solar interior conditions. We discuss some theoretical aspects such as density effects, photo-ionization, autoionization or the "filling-the-gap" effect of highly excited states.
We will calculate the essential dimension of the moduli stack of polarized K3 surfaces, including the positive char case.
The basic equations describing a Newtonian universe with uniform pressure are reexamined. We argue that in the semi-classical formulation adopted in the literature the continuity equation has a misleading pressure gradient term. When this term is removed, the resulting equations give the same homogeneous background solutions with pressure and the same evolution equation for the density contrast as are obtained using the full relativistic approach.
Recently, we have established the generalized Li criterion equivalent to the Riemann hypothesis, viz. demonstrated that the sums over all non-trivial Riemann function zeroes k_n,a=Sum_(/rho)(1-(1-((/rho-a)/(/rho+a-1))^n) for any real a not equal to 1/2 are non-negative if and only if the Riemann hypothesis holds true, and proved the relation k_n,a=n*(1-2a)/(n-1)!*d^n/dz^n((z-a)^(n-1)*ln(\xi(z))) taken at z=1-a. Assuming that the function /zeta(s) is non-vanishing for Re(s)>1/2+/Delta, where real 0</Delta<1/2, using this relation together with the functional equation for the /xi-function and the explicit formula of Weil, we prove that in these conditions for n=1, 2, 3... and an arbitrary complex a with 1>Re(a)>1/2+/Delta+delta_0, where /delta_0 is an arbitrary small fixed positive number, one has d^n/ds^n(ln(/zeta(s))=Sum_(m<=N)((-1)^n*/Lambda(m)*ln^(n-1)(m)/m^a) + Int_(0)^(N)(x^(-a)*ln^(n-1)(x)*dx)+O(N^(1/2+Delta-a)*ln^(n-1)(N)); derivative is taken at s=a. In particular, d(ln(/zeta(a))/da=-Sum_(m<=N)(/Lambda(m)/m^a+N^(1-a)/(1-a)+O(N^(1/2+/Delta-a)). Numerical verifications of these equalities are also presented.
The quantum gravity path integral's measure can be written as the product of classical backgrounds and quantum fluctuations about each background. After proving that fluctuations about the background do not diffuse in Hilbert space and obey the laws of many-body statistics, their probability distributions, entropy, and expected background are determined. This background obeys expectation-valued Einstein equations and features an entropy-based positive cosmological constant. From the fluctuation probability distributions, a finite temperature, R-squared, quantum gravity path integral is constructed whose action presents an interaction picture of quantum gravity that `moves with' the expected background in Hilbert space. Within this interaction picture of quantum fluctuations about an expected background, the fields required to describe quantum gravity have been transformed into `ordinary' quantum fields propagating on this `rigid' or `fixed' expected background. Back-reaction has been fully accounted for, and the quantum formulation is manifestly background independent.
In a recent work, Andrews gave analytic proofs of two conjectures concerning some variations of two combinatorial identities between partitions of a positive integer into odd parts and partitions into distinct parts discovered by Beck. Subsequently, using the same method as Andrews, Chern presented the analytic proof of another Beck's conjecture relating the gap-free partitions and distinct partitions with odd length. However, the combinatorial interpretations of these conjectures are still unclear and required. In this paper, motivated by Glaisher's bijection, we give the combinatorial proofs of these three conjectures directly or by proving more generalized results.
These are notes of lectures given at the school `Birational Geometry of Hypersurfaces' in Gargnano in March 2018. The main goal was to discuss the Hodge structures that come naturally associated with a cubic fourfold. The emphasis is on the Hodge and lattice theoretic aspects with many technical details worked out explicitly. More geometric or derived results are only hinted at.
We prove the central limit theorem for the volume and the $f$-vector of the Poisson random polytope $\Pi_{\eta}$ in a fixed convex polytope $P\subset\mathbb{R}^d$. Here, $\Pi_{\eta}$ is the convex hull of the intersection of a Poisson process $X$ of intensity $\eta$ with $P$.