ShantanuT01/DACTYL-Pretraining huggingface dataset:
We consider Schroedinger operators on metric cones whose cross section is a closed Riemannian manifold $(Y, h)$ of dimension $d-1 \geq 2$. Thus the metric on the cone $M = (0, \infty)_r \times Y$ is $dr^2 + r^2 h$. Let $\Delta$ be the Friedrichs Laplacian on $M$ and $V_0$ be a smooth function on $Y$, such that $\Delta_Y + V_0 + (d-2)^2/4$ is a strictly positive operator on $L^2(Y)$, with lowest eigenvalue $\mu^2_0 $ and second lowest eigenvalue $\mu^2_1$, with $\mu_0, \mu_1 > 0$. The operator we consider is $H = \Delta + V_0/r^2$, a Schr\"odinger operator with inverse square potential on $M$; notice that $H$ is homogeneous of degree -2.   We study the Riesz transform $T = \nabla H^{-1/2}$ and determine the precise range of $p$ for which $T$ is bounded on $L^p(M)$. This is achieved by making a precise analysis of the operator $(H + 1)^{-1}$ and determining the complete asymptotics of its integral kernel. We prove that if $V$ is not identically zero, then the range of $p$ for $L^p$ boundedness is   $$ d/ \Big(min(1+d/2+\mu_0, d \Big) < p < d / \Big(max(d/2-\mu_0, 0) \Big),$$ while if $V$ is identically zero, then the range is   $$ 1 < p < d / \Big(max(d/2-\mu_1, 0 \Big).$$   The result in the case $V$ identically zero was first obtained in a paper by H.-Q. Li.
We study in a semiclassical regime a two-dimensional magnetic periodic Schr\"odinger operator. We first review some results for the square (Harper), triangular and hexagonal (case of the graphene) lattices. Then we study the case considered by Hou when the periodicity is given by a kagome lattice. We reduce the problem to the study of discrete and pseudodifferential operators and obtain pictures similar to Hofstadter's butterfly. We prove the existence of flat bands, which do not occur in the three previous cases.
Using a recently proposed nonlinear force-free method designed for single vector magnetograms of solar active regions we calculate the instantaneous free magnetic energy and relative magnetic helicity budgets in 162 vector magnetograms corresponding to 42 different active regions. We find a statistically robust, monotonic correlation between the free magnetic energy and the relative magnetic helicity in the studied regions. This correlation implies that magnetic helicity, besides free magnetic energy, may be an essential ingredient for major solar eruptions. Eruptive active regions appear well segregated from non-eruptive ones in both free energy and relative helicity with major (at least M-class) flares occurring in active regions with free energy and relative helicity exceeding 4x10^{31} erg and 2x10^{42} Mx^2, respectively. The helicity threshold agrees well with estimates of helicity contents of typical coronal mass ejections.
We consider the Cauchy problem of a class of higher order Schr\"odinger type equations with constant coefficients. By employing the energy inequality, we show the $L^2$ well-posedness, the parabolic smoothing and a breakdown of the persistence of regularity. We classify this class of equations into three types on the basis of their smoothing property.
In this paper we define and study a new family of graphs that generalises the notions of line graphs and path graphs. Let $G$ be a graph with no loops but possibly with parallel edges. An \emph{$\ell$-link} of $G$ is a walk of $G$ of length $\ell \geqslant 0$ in which consecutive edges are different. We identify an $\ell$-link with its reverse sequence. The \emph{$\ell$-link graph $\mathbb{L}_\ell(G)$} of $G$ is the graph with vertices the $\ell$-links of $G$, such that two vertices are joined by $\mu \geqslant 0$ edges in $\mathbb{L}_\ell(G)$ if they correspond to two subsequences of each of $\mu$ $(\ell + 1)$-links of $G$.   By revealing a recursive structure, we bound from above the chromatic number of $\ell$-link graphs. As a corollary, for a given graph $G$ and large enough $\ell$, $\mathbb{L}_\ell(G)$ is $3$-colourable. By investigating the shunting of $\ell$-links in $G$, we show that the Hadwiger number of a nonempty $\mathbb{L}_\ell(G)$ is greater or equal to that of $G$. Hadwiger's conjecture states that the Hadwiger number of a graph is at least the chromatic number of that graph. The conjecture has been proved by Reed and Seymour (2004) for line graphs, and hence $1$-link graphs. We prove the conjecture for a wide class of $\ell$-link graphs.
A local interpretation of quantum mechanics is presented. Its main ingredients are: first, a label attached to one of the virtual paths in the path integral formalism, determining the output for measurement of position or momentum; second, a mathematical model for spin states, equivalent to the path integral formalism for point particles in space time, with the corresponding label. The mathematical machinery of orthodox quantum mechanics is maintained, in particular amplitudes of probability and Born's rule; therefore, Bell's type inequalities theorems do not apply. It is shown that statistical correlations for pairs of particles with entangled spins have a description completely equivalent to the two slit experiment, that is, interference (wave like behaviour) instead of non locality gives account of the process. The interpretation is grounded in the experimental evidence of a point like character of electrons, and in the hypothetical existence of a wave like, the de Broglie, companion system. A correspondence between the extended Hilbert spaces of hidden physical states and the orthodox quantum mechanical Hilbert space shows the mathematical equivalence of both theories. Paradoxical behaviour with respect to the action reaction principle is analysed, and an experimental set up, modified two slit experiment, proposed to look for the companion system.
We have used the de Hass-van Alphen (dHvA) effect to investigate the Fermi surface of high-quality crystalline TiSb2, which unveiled a nontrivial topologic nature by analyzing the dHvA quantum oscillations. Moreover, our analysis on the quantum oscillation frequencies associated with nonzero Berry phase when the magnetic field is parallel to both of the ab-plane and c-axis of TiSb2 finds that the Fermi surface topology has a three-dimensional (3D) feature. The results are supported by the first-principle calculations which revealed a symmetry-protected Dirac point appeared along the {\Gamma}-Z high symmetry line near the Fermi level. On the (001) surface, the bulk Dirac points are found to project onto the -{\Gamma} point with nontrivial surface states. Our finding will substantially enrich the family of 3D Dirac semimetals which are useful for topological applications.
Supergiant High Mass X-ray Binary systems (sgHMXBs) consist of a massive, late type, star and a neutron star. The massive stars exhibits strong, radiatively driven, stellar winds. Wind accretion onto compact object triggers X-ray emission, which alters the stellar wind significantly. Hydrodynamic simulation has been used to study the neutron star - stellar wind interaction it two sgHMXBs: i) A heavily obscured sgHMXB (IGR J17252-3616) discovered by INTEGRAL. To account for observable quantities (i.e., absorbing column density) we have to assume a very slow wind terminal velocity of about 500 km/s and a rather massive neutron star. If confirmed in other obscured systems, this could provide a completely new stellar wind diagnostics. ii) A classical sgHMXB (Vela X-1) has been studied in depth to understand the origin of the off-states observed in this system. Among many models used to account for this observed behavior (clumpy wind, gating mechanism) we propose that self-organized criticality of the accretion stream is the likely reason for the observed behavior. In conclusion, the neutron star, in these two examples, acts very efficiently as a probe to study stellar winds.
In this paper we continue our research on the concept of liken. This notion has been defined as a sequence of non-negative real numbers, tending to infinity and closed with respect to addition in $\mathbb{R}$. The most important examples of likens are clearly the set of natural numbers $\mathbb{N}$ with addition and the set of positive natural numbers $\mathbb{N}^{*}$ with multiplication, represented by a sequence $(\ln({n+1}))_0^{\infty}$.   The set of all likens can be parameterized by the points of some infinite dimensional, complete metric space. In this space of likens we consider elements up to isomorphism and define properties of likens as such, that are isomorphism invariant. The main result of this work is a theorem characterizing the liken ${\mathbb N}^*$ of natural numbers with multiplication in the space of all likens.
Let $c_1(x),c_2(x),f_1(x),f_2(x)$ be polynomials with rational coefficients. With obvious exceptions, there can be at most finitely many roots of unity among the zeros of the polynomials $c_1(x)f_1(x)^n+c_2(x)f_2(x)^n$ with $n=1,2\ldots$. We estimate the orders of these roots of unity in terms of the degrees and the heights of the polynomials $c_i$ and $f_i$.
We study analytically a simple game theoretical model of heterogeneous interacting agents. We show that the stationary state of the system is described by the ground state of a disordered spin model which is exactly solvable within the simple replica symmetric ansatz. Such a stationary state differs from the Nash equilibrium where each agent maximizes her own utility. The latter turns out to be characterized by a replica symmetry broken structure. Numerical results fully agree with our analytic findings.
In the past few decades, observations indicated that an unexplained high production rate of positrons (the strong 511 keV line) exists in the Milky Way center. By using the fact that a large amount of high density gas used to exist near the Milky Way center million years ago, we model the rate of positrons produced due to dark matter annihilation. We consider the effect of adiabatic contraction of dark matter density due to the supermassive black hole at the Milky Way center and perform a detailed calculation to constrain the possible annihilation channel and dark matter mass range. We find that only three annihilation channels ($\mu^+\mu^-$, $4e$ and $4 \mu$) can provide the required positron production rate and satisfy the stringent constraint of gamma-ray observations. In particular, the constrained mass range for the $\mu^+\mu^-$ channel is $m \approx 80-100$ GeV, which is close to the mass range obtained for the dark matter interpretation of the GeV gamma-ray and positron excess. In other words, the proposed scenario can simultaneously provide the required positron production rate to explain the 511 keV emission, the positron excess and the GeV gamma-ray excess in our Milky Way, and it is compatible with the density spike due to adiabatic growth model of the supermassive black hole.
In the framework of schematic hard spheres lattice models for granular media we investigate the phenomenon of the ``jamming transition''. In particular, using Edwards' approach, by analytical calculations at a mean field level, we derive the system phase diagram and show that ``jamming'' corresponds to a phase transition from a ``fluid'' to a ``glassy'' phase, observed when crystallization is avoided. Interestingly, the nature of such a ``glassy'' phase turns out to be the same found in mean field models for glass formers.
Based on the dielectric continuum model, we calculated the phonon assisted tunneling (PAT) current of general double barrier resonant tunneling structures (DBRTSs) including both symmetric and antisymmetric ones. The results indicate that the four higher frequency interface phonon modes (especially the one which peaks at either interface of the emitter barrier) dominate the PAT processes, which increase the valley current and decrease the PVR of the DBRTSs. We show that an asymmetric structure can lead to improved performance.
Just like the vector gauge bosons in the gauge theories, it is now known that gravitons acquire mass in the process of spontaneous symmetry breaking of diffeomorphisms through the condensation of scalar fields. The point is that we should find the gravitational Higgs mechanism such that it results in massive gravity in a flat Minkowski space-time without non-unitary propagating modes. This is usually achieved by including higher-derivative terms in scalars and tuning the cosmological constant to be a negative value in a proper way. Recently, a similar but different gravitational Higgs mechanism has been advocated by Chamseddine and Mukhanov where one can relax the negative cosmological constant to zero or positive one. In this work, we investigate why the non-unitary ghost mode decouples from physical Hilbert space in a general space-time dimension. Moreover, we generalize the model to possess an arbitrary potential and clarify under what conditions the general model exhibits the gravitational Higgs mechanism. By searching for solutions to the conditions, we arrive at two classes of potentials exhibiting gravitational Higgs mechanism. One class includes the model by Chamseddine and Mukhanov in a specific case while the other is completely a new model.
More than 40 years of ground-based photometric observations of the delta Sct star 4CVn revealed 18 independent oscillation frequencies, including radial as well as non-radial p-modes of low spherical degree l<=2. From 2008 to 2011, more than 2000 spectra were obtained at the 2.1-m Otto-Struve telescope at the McDonald Observatory. We present the analysis of the line-profile variations, based on the Fourier-parameter fit method, detected in the absorption lines of 4CVn, which carry clear signatures of the pulsations. From a non-sinusoidal, periodic variation of the radial velocities, we discovered that 4CVn is an eccentric binary system, with an orbital period Porb = 124.44 +/- 0.03 d and an eccentricity e = 0.311 +/- 0.003. We firmly detect 20 oscillation frequencies, 9 of which are previously unseen in photometric data, and attempt mode identification for the two dominant modes, f1 = 7.3764 c/d and f2 = 5.8496 c/d, and determine the prograde or retrograde nature of 7 of the modes. The projected rotational velocity of the star, vsini ~ 106.7 km/s, translates to a rotation rate of veq/vcrit >= 33%. This relatively high rotation rate hampers unique mode identification, since higher-order effects of rotation are not included in the current methodology. We conclude that, in order to achieve unambiguous mode identification for 4CVn, a complete description of rotation and the use of blended lines have to be included in mode-identification techniques.
In this paper, we prove the following "Weak Bounded Negativity Conjecture", which says that given a complex smooth projective surface $X$, for any reduced curve $C$ in $X$ and integer $g$, assume that the geometric genus of each component of $C$ is bounded from above by $g$, then the self-intersection number $C^2$ is bounded from below.
We present a visibility based estimator namely, the Tapered Gridded Estimator (TGE) to estimate the power spectrum of the diffuse sky signal. The TGE has three novel features. First, the estimator uses gridded visibilities to estimate the power spectrum which is computationally much faster than individually correlating the visibilities. Second, a positive noise bias is removed by subtracting the auto-correlation of the visibilities which is responsible for the noise bias. Third, the estimator allows us to taper the field of view so as to suppress the contribution from the sources in the outer regions and the sidelobes of the telescope's primary beam. We first consider the two dimensional (2D) TGE to estimate the angular power spectrum $C_{\ell}$. We have also extended the TGE to estimate the three dimensional (3D) power spectrum $P({\bf k})$ of the cosmological 21-cm signal. Analytic formulas are presented for predicting the variance of the binned power spectrum. Both the estimators and their variance predictions are validated using simulations of $150 \, {\rm MHz}$ GMRT observations. We have applied the 2D TGE to estimate $C_{\ell}$ using visibility data for two of the fields observed by TIFR GMRT Sky Survey (TGSS). We find that the sky signal, after subtracting the point sources, is likely dominated by the diffuse Galactic synchrotron radiation across the angular multipole range $240 \le \ell \lesssim 500$.
Time evolution of wave packets built from the eigenstates of the Dirac equation for a hydrogenic system is considered. We investigate the space and spin motion of wave packets which, in the non-relativistic limit, are stationary states with a probability density distributed uniformly along the classical, elliptical orbit (elliptic WP). We show that the precession of such a WP, due to relativistic corrections to the energy eigenvalues, is strongly correlated with the spin motion. We show also that the motion is universal for all hydrogenic systems with an arbitrary value of the atomic number Z.
The review paper presents generalization of d'Alembert's variational principle: the dynamics of a quantum system for an external observer is defined by the exact equilibrium of all acting in the system forces, including the random quantum force $\hbar j$, $\forall\hbar$. Spatial attention is dedicated to the systems with (hidden) symmetries. It is shown how the symmetry reduces the number of quantum degrees of freedom down to the independent ones. The sin-Gordon model is considered as an example of a such field theory with symmetry. It is shown why the particles $S$-matrix is trivial in that model.
We study the importance of gravitational lensing in the modelling of the number counts of galaxies. We confirm previous results for photometric surveys, showing that lensing cannot be neglected in a survey like LSST since it would infer a significant shift of cosmological parameters. For a spectroscopic survey like SKA2, we find that neglecting lensing in the monopole, quadrupole and hexadecapole of the correlation function also induces an important shift of parameters. For ${\Lambda}$CDM parameters, the shift is moderate, of the order of 0.6${\sigma}$ or less. However, for a model-independent analysis, that measures the growth rate of structure in each redshift bin, neglecting lensing introduces a shift of up to 2.3${\sigma}$ at high redshift. Since the growth rate is directly used to test the theory of gravity, such a strong shift would wrongly be interpreted as the breakdown of General Relativity. This shows the importance of including lensing in the analysis of future surveys. On the other hand, for a survey like DESI, we find that lensing is not important, mainly due to the value of the magnification bias parameter of DESI, $s(z)$, which strongly reduces the lensing contribution at high redshift. We also propose a way of improving the analysis of spectroscopic surveys, by including the cross-correlations between different redshift bins (which is neglected in spectroscopic surveys) from the spectroscopic survey or from a different photometric sample. We show that including the cross-correlations in the SKA2 analysis does not improve the constraints. On the other hand replacing the cross-correlations from SKA2 by cross-correlations measured with LSST improves the constraints by 10 to 20 %. Interestingly, for ${\Lambda}$CDM parameters, we find that LSST and SKA2 are highly complementary, since they are affected differently by degeneracies between parameters.
We study graded rings associated to big divisors on LC pairs whose difference with the log-canonical divisor is nef. For divisors that are positive enough at the LC centers of the pair, we prove the finite generation of such rings if the pair is DLT or the dimension is low, given that a Zariski decomposition exists.
Linear [Q/H]-[O/H] relations, Q = Na, Mg, Si, Ca, Ti, Cr, Fe, Ni, are inferred from a sample (N=67) of recently studied FGK-type dwarf stars in the solar neighbourhood including different populations. Regression line slope and intercept estimators and related variance estimators are determined. With regard to the straight line, [Q/H] = aQ [O/H] + bQ, sample stars display along a "main sequence", [Q,O] = [aQ,bQ, Delta bQ], leaving aside the two OL stars which, in most cases (e.g., Na), lie outside. A unit slope, aQ = 1, implies Q is a primary element synthesised via SNII progenitors in presence of universal stellar initial mass function (defined as simple primary element). To this respect, Mg, Si, Ti, show aQ = 1 within 2 sigma(aQ); Cr, Fe, Ni, within 3 sigma(aQ); Na, Ca, exceeding 3 sigma(aQ). The empirical, differential element abundance distributions are inferred from different subsamples, where related regression lines represent their theoretical counterparts within the framework of simple MCBR (multistage closed box + reservoir) chemical evolution models. Hence the fractional yields are determined and (as an example) a comparison is shown with their theoretical counterparts inferred from SNII progenitor nucleosynthesis under the assumption of a power-law stellar initial mass function. The generalized fractional yields are determined regardless from the chemical evolution model. The ratio of outflow to star formation rate is compared for different populations, in the framework of simple MCBR models. The opposite situation of element abundance variation entirely due to cosmic scatter is also considered under reasonable assumptions. The related differential element abundance distribution fits to the data as well as its counterpart inferred in the opposite limit of instantaneous mixing in presence of chemical evolution.
We present here a new calculation of the gamma-ray spectrum from pp->pi^0 in the Galactic ridge environment. The calculation includes the diffractive pp interaction and incorporates the Feynman scaling violation for the first time. Galactic diffuse gamma-rays come, predominantly, from pi^0->gamma gamma in the sub-GeV to multi-GeV range. Hunter et al. found, however, an excess in the GeV range ("GeV Excess") in the EGRET Galactic diffuse spectrum above the prediction based on experimental pp->pi^0 cross-sections and the Feynman scaling hypothesis. We show, in this work, that the diffractive process makes the gamma-ray spectrum harder than the incident proton spectrum by ~0.05 in power-law index, and, that the scaling violation produces 30-80% more pi^0 than the scaling model for incident proton energies above 100GeV. Combination of the two can explain about a half of the "GeV Excess" with the local cosmic proton (power-law index ~2.7). The excess can be fully explained if the proton spectral index in the Galactic ridge is a little harder (~0.2 in power-law index) than the local spectrum. Given also in the paper is that the diffractive process enhances e^+ over e^- and the scaling violation gives 50-100% higher p-bar yield than without the violation, both in the multi-GeV range.
HESS J1731$-$347 a.k.a. SNR G353.6$-$0.7 is one of the five known shell-type supernova remnants (SNRs) emitting in the very high energy (VHE, Energy $>$ 0.1 TeV) $\gamma$-ray domain. We observed this TeV SNR with the Giant Metrewave Radio Telescope (GMRT) in 1390, 610 and 325 MHz bands. In this paper, we report the discovery of 325 and 610 MHz radio counterparts of the SNR HESS J1731$-$347 with the GMRT. Various filaments of the SNR are clearly seen in the 325 and 610 MHz bands. However, the faintest feature in the radio bands corresponds to the peak in VHE emission. We explain this anti-correlation in terms of a possible leptonic origin of the observed VHE $\gamma$-ray emission. We determine the spectral indices of the bright individual filaments, which were detected in both the 610 and the 325 MHz bands. Our values range from $-$1.11 to $-$0.15, consistent with the non-thermal radio emission. We also report a possible radio counterpart of a nearby TeV source HESS J1729$-$345 from the 843 MHz Molonglo Galactic Plane Survey and the 1.4 GHz Southern Galactic Plane Survey maps. The positive radio spectral index of this possible counterpart suggests a thermal origin of the radio emission of this nearby TeV source.