ShantanuT01/DACTYL-Pretraining huggingface dataset:
We discuss how luminosity and space distribution of galaxies are naturally linked in view of their multifractal properties. In particular we show that the mass (luminosity) function corresponding to a multifractal distribution in a given observed volume, consists of a power law followed by an exponential cut-off. This implies that the amplitude $\phi^*$ of the Schechter function scales with the sample depth, as confirmed by various observational data. This effect is analogous to the scaling of the space density due to the fractal nature of the space distribution. This has the important consequence that the luminosity function can be properly defined only in a volume limited sample. Also the so-called "luminosity segregation" and the concept of bias correspond to a natural consequence of multifractality. This implies however that they should be considered from a different perspective with respect to the usual one. Such a concept allows us to unify the space and luminosity distributions as being shaped by a single cause:multifractality which should therefore claim a central stage in theoretical investigations.
Our goal is to find a minimalistic model that describes the luminosity function and large-scale clustering bias of the X-ray-selected AGN in the general framework of the concordance LCDM model. We assume that a simple population-averaged scaling relation between the AGN X-ray luminosity L_X and the host dark matter halo mass M_h exists. With such a relation, the AGN X-ray luminosity function can be computed from the halo mass function. Using the concordance LCDM halo mass function for the latter, we obtain the M_h-L_X relation required to match the redshift-dependent AGN X-ray luminosity function known from X-ray observations. We find that with a simple power-law-scaling M_h \propto L_X^\Gamma(z), our model can successfully reproduce the observed X-ray luminosity function. Furthermore, we automatically obtain predictions for the large-scale AGN clustering amplitudes and their dependence on the luminosity and redshift, which seem to be compatible with AGN clustering measurements. Our model also includes the redshift-dependent AGN duty cycle which peaks at the redshift z ~ 1, and its peak value is consistent with unity, suggesting that on average there is no more than one AGN per dark matter halo. For a typical X-ray-selected AGN at z ~ 1, our best-fit M_h-L_X scaling implies low Eddington ratio L_X/L_Edd ~ 10^{-4}-10^{-3} (2-10 keV band, no bolometric correction applied) and correspondingly large mass growth e-folding times, suggesting that the typical X-ray AGN are dominantly fueled via relatively inefficient 'hot-halo' accretion mode.
This article is devoted to the construction of numerical methods which remain insensitive to the smallness of the semiclassical parameter for the linear Schr{\"o}dinger equation in the semiclassical limit. We specifically analyse the convergence behavior of the first-order splitting. Our main result is a proof of uniform accuracy. We illustrate the properties of our methods with simulations.
Free boundary problems are those described by PDEs that exhibit a priori unknown (free) interfaces or boundaries. These problems appear in Physics, Probability, Biology, Finance, or Industry, and the study of solutions and free boundaries uses methods from PDEs, Calculus of Variations, Geometric Measure Theory, and Harmonic Analysis. The most important mathematical challenge in this context is to understand the structure and regularity of free boundaries.   In this paper we provide an invitation to this area of research by presenting, in a completely non-technical manner, some classical results as well as some recent results of the author.
We study the higher spin anologs of the six vertex model on the basis of its symmetry under the quantum affine algebra $U_q(\slth)$. Using the method developed recently for the XXZ spin chain, we formulate the space of states, transfer matrix, vacuum, creation/annihilation operators of particles, and local operators, purely in the language of representation theory. We find that, regardless of the level of the representation involved, the particles have spin $1/2$, and that the $n$-particle space has an RSOS-type structure rather than a simple tensor product of the $1$-particle space. This agrees with the picture proposed earlier by Reshetikhin.
Shor's algorithm contains a classical post-processing part for which we aim to create an efficient, understandable method aside from continued fractions.   Let r be an unknown positive integer. Assume that with some constant probability we obtain random positive integers of the form x=[ N k/r ] where [.] is either the floor or ceiling of the rational number, k is selected uniformly at random from {0,1,...,r-1}, and N is a parameter that can be chosen. The problem of recovering r from such samples occurs precisely in the classical post-processing part of Shor's algorithm. The quantum part (quantum phase estimation) makes it possible to obtain such samples where r is the order of some element a of the unit group of Z_n and n is the number to be factored.   Shor showed that the continued fraction algorithm can be used to efficiently recover r, since if N>2r^2 then k/r appears in lowest terms as one of the convergents of x/N due to a standard result on continued fractions. We present here an alternative method for recovering r based on the Gauss algorithm for lattice basis reduction, allowing us to efficiently find the shortest nonzero vector of a lattice generated by two vectors. Our method is about as efficient as the method based on continued fractions, yet it is much easier to understand all the details of why it works.
The methods of density-functional perturbation theory may be used to calculate various physical response properties of insulating crystals including elastic, dielectric, Born charge, and piezoelectric tensors. These and other important tensors may be defined as second derivatives of the total energy with respect to atomic-displacement, electric-field, or strain perturbations, or as mixed derivatives with respect to two of these perturbations. The resulting tensor quantities tend to be coupled in complex ways in polar crystals, giving rise to a variety of variant definitions. For example, it is generally necessary to distinguish between elastic tensors defined under different electrostatic boundary conditions, and between dielectric tensors defined under different elastic boundary conditions. Here, we describe an approach for computing all of these various response tensors in a unified and systematic fashion. Applications are presented for two materials, wurtzite ZnO and rhombohedral BaTiO3, at zero temperature.
We use stellar dynamics, strong lensing, stellar population synthesis models, and weak lensing shear measurements to constrain the dark matter (DM) profile and stellar mass in a sample of 53 massive early-type galaxies. We explore three DM halo models (unperturbed Navarro Frenk & White [NFW] halos and the adiabatic contraction models of Blumenthal and Gnedin) and impose a model for the relationship between the stellar and virial mass (i.e., a relationship for the star-formation efficiency as a function of halo mass). We show that, given our model assumptions, the data clearly prefer a Salpeter-like initial mass function (IMF) over a lighter IMF (e.g., Chabrier or Kroupa), irrespective of the choice of DM halo. In addition, we find that the data prefer at most a moderate amount of adiabatic contraction (Blumenthal adiabatic contraction is strongly disfavored) and are only consistent with no adiabatic contraction (i.e., a NFW halo) if a mass-dependent IMF is assumed, in the sense of a more massive normalization of the IMF for more massive halos.
A now classical construction due to Kato and Nakayama attaches a topological space (the "Betti realization") to a log scheme over $\mathbf{C}$. We show that in the case of a log smooth degeneration over the standard log disc, this construction allows one to recover the topology of the germ of the family from the log special fiber alone. We go on to give combinatorial formulas for the monodromy and the $d^2$ differentials acting on the nearby cycle complex in terms of the log structures. We also provide variants of these results for the Kummer etale topology. In the case of curves, these data are essentially equivalent to those encoded by the dual graph of a semistable degeneration, including the monodromy pairing and the Picard-Lefschetz formula.
Magnetization and heat capacity measurements of ternary rare earth intermetallic compound GdNiAl3 demonstrate para to ferromagnetic transition at Tc=165.5K. In addition multiple short range magnetic transitions observed below Tc are suggestive of competing interactions in this compound. As a result of this a weak Griffiths phase type behaviour is observed in the paramagnetic region. This complex behaviour is rather supported by the random orientation of Ni centered tricapped trigonal prisms with additional Al atoms in the structure. Heat capacity and resistivity data display an interesting peak at 72 K, which is highly unaffected by magnetic fields up to 90KOe.
We investigate a graph theoretic analog of geodesic geometry. In a graph $G=(V,E)$ we consider a system of paths $\mathcal{P}=\{P_{u,v}|u,v\in V\}$ where $P_{u,v}$ connects vertices $u$ and $v$. This system is consistent in that if vertices $y, z$ are in $P_{u,v}$, then the sub-path of $P_{u,v}$ between them coincides with $P_{y,z}$. A map $w: E\to(0,\infty)$ is said to induce $\mathcal{P}$ if for every $u, v\in V$ the path $P_{u,v}$ is $w$-geodesic. We say that $G$ is metrizable if every consistent path system is induced by some such $w$. As we show, metrizable graphs are very rare, whereas there exist infinitely many $2$-connected metrizable graphs.
We show that a cone theorem for ${\mathbbA}^1-homotopy invariant contravariant functors implies the vanishing of the positive degree part of the operational Chow cohomology rings of a large class of affine varieties. We also discuss how this vanishing relates to a number of questions about representing Chow cohomology classes of GIT quotients in terms of equivariant cycles.
The limits of the areal storage density as can be achieved with heat assisted magnetic recording (HAMR) are still an open issue. We want to address this central question and present the design of a possible bit patterned medium with an areal storage density above 10 Tb/in$^2$. The model uses hard magnetic recording grains with 5 nm diameter and 10 nm height. It assumes a realistic distribution of the Curie temperature of the underlying material as well as a realistic distribution of the grain size and the grain position. In order to compute the areal density we analyze the detailed switching behavior of a recording bit under different external conditions, which allows to compute the bit error rate of a recording process (shingled and conventional) for different grain spacings and write head positions. Hence, we are able to optimize the areal density of the presented medium.
Many hot subdwarf B stars (sdBs) are in close binaries, and the favored formation channels for subdwarfs rely on mass transfer in a binary system to strip a core He burning star of its envelope. However, these channels cannot account for sdBs that have been observed in long period binaries nor the narrow mass distribution of isolated (or "singleton") sdBs. We propose a new formation channel involving the merger of a helium white dwarf and a low mass, hydrogen burning star, which addresses these issues. Hierarchical triples whose inner binaries merge and form sdBs by this process could explain the observed long period subdwarf+main sequence binaries. This process would also naturally explain the observed slow rotational speeds of singleton sdBs. We also briefly discuss the implications of this formation channel for extreme horizontal branch morphology in globular clusters and the UV upturn in elliptical galaxies.
A multiset hook length formula for integer partitions is established by using combinatorial manipulation. As special cases, we rederive three hook length formulas, two of them obtained by Nekrasov-Okounkov, the third one by Iqbal, Nazir, Raza and Saleem, who have made use of the cyclic symmetry of the topological vertex. A multiset hook-content formula is also proved.
We calculate thermal fluctuation properties: volume-averaged order parameter, Helmholtz free and internal energies, and their variances of a supersaturated disordered phase in the Gibbs canonical ensemble for an asymmetric (third-order interactions), athermal (independence of the supersaturation and thermal noise) effective Hamiltonian. These properties are different from those of the symmetric thermal one with the most important differences being the phase coexistence and "thermal expansion." The fluctuation properties of the system were calculated theoretically, using the perturbation method, and numerically, using the "brute force" simulations method. Overall, the numerical calculations match the theory within the accuracy of the numerical method. However, a discrepancy of the dependence of the internal energy and its variance on the supersaturation exists. Results of the present study can be used for calculations of the fluctuation properties of the systems and modeling of nucleation and other rare events in the framework of the Ginzburg-Landau method.
We study the optical absorption spectra in a polar ferrimagnet GaFeO_3. We consider the E1, E2 and M1 processes on Fe atoms. It is shown that the magnetoelectric effect on the absorption spectra arises from the E1-M1 interference process through the hybridization between the 4p and 3d states in the noncentrosymmetry environment of Fe atoms. We perform a microscopic calculation of the spectra on a cluster model of FeO_6 consisting of an octahedron of O atoms and an Fe atom displaced from the center with reasonable values for Coulomb interaction and hybridization. We obtain the magnetoelectric spectra, which depend on the direction of magnetization, as a function of photon energy in the optical region 1.0-2.5 eV, in agreement with the experiment.
We introduce the concept of strong quantum speedup. We prove that approximating the ground state energy of an instance of the time-independent Schr\"odinger equation, with $d$ degrees of freedom, $d$ large, enjoys strong exponential quantum speedup. It can be easily solved on a quantum computer. Some researchers in discrete complexity theory believe that quantum computation is not effective for eigenvalue problems. One of our goals in this paper is to explain this dissonance.
We explore the properties of the H_beta emission line profile in a large, homogeneous and bright sample of N~470 low redshift quasars extracted from Sloan Digital Sky Survey (DR5). We approach the investigation from two complementary directions: composite/median spectra and a set of line diagnostic measures (asymmetry index, centroid shift and kurtosis) in individual quasars. The project is developed and presented in the framework of the so-called 4D Eigenvector 1 (4DE1) Parameter Space, with a focus on its optical dimensions, FWHM(H_beta) and the relative strength of optical FeII (R_FeII=W(FeII4434-4684)/W(H_beta)). We reenforce the conclusion that not all quasars are alike and spectroscopically they do not distribute randomly about an average typical optical spectrum. Our results give further support to the concept of two populations A and B (narrower and broader than 4000 km/s FWHM(H_beta), respectively) that emerged in the context of 4DE1 space. The broad H_beta profiles in composite spectra of Population A sources are best described by a Lorentzian and in Population B by a double Gaussian model. Moreover, high and low accretion sources (an alternative view of the Population A/B concept) not only show significant differences in terms of Black Hole (BH) mass and Eddington ratio L_bol/L_Edd, but they also show distinct properties in terms of line asymmetry, shift and shapes. We finally suggest that a potential refinement of the 4DE1 space can be provided by separating two populations of quasars at R_FeII~0.50 rather than at FWHM(H_beta)=4000 km/s. Concomitantly, the asymmetry and centroid shift profile measures at 1/4 fractional intensity can be reasonable surrogates for the FWHM(H_beta) dimension of the current 4DE1.
We argue that the QCD matter not far above a critical confinement-deconfinement baryon density and low temperatures can develop spontaneously the condensates of spin-one quark Cooper pairs. Depending upon their color these condensates characterize two distinct anisotropic color-superconducting phases. For them we derive the generic form of the quasiquark dispersion laws and the gap equation. We also visualize the soft Nambu-Goldstone modes of spontaneously broken global symmetries, and demonstrate an unusual form of the Meissner effect.}]
Aims: We generate theoretical ultraviolet and extreme-ultraviolet emission line ratios for O IV and show their strong versatility as electron temperature and density diagnostics for astrophysical plasmas. Methods: Recent fully relativistic calculations of radiative rates and electron impact excitation cross sections for O IV, supplemented with earlier data for A-values and proton excitation rates, are used to derive theoretical O IV line intensity ratios for a wide range of electron temperatures and densities. Results: Diagnostic line ratios involving ultraviolet or extreme-ultraviolet transitions in O IV are presented, that are applicable to a wide variety of astrophysical plasmas ranging from low density gaseous nebulae to the densest solar and stellar flares. Comparisons with observational data, where available, show good agreement between theory and experiment, providing support for the accuracy of the diagnostics. However, diagnostics are also presented involving lines that are blended in existing astronomical spectra, in the hope this might encourage further observational studies at higher spectral resolution.
We explore the possible occurrence of $\sigma$-$\omega$ condensation in the Quantum Hadro-Dynamics (QHD), namely the Serot and Walecka model, finding that at the mean field level it corresponds to a critical value of the coupling constant $g_\sigma=8.828$ and density $k_F=207.2$ MeV/c, significantly below the standard value of QHD.
We consider Navier-Stokes equations for compressible viscous fluids in one dimension. We prove the existence of global strong solution with large initial data for the shallow water system. The key ingredient of the proof relies to a new formulation of the compressible equations involving a new effective velocity $v$ (see \cite{cras,para,CPAM,CPAM1}) such that the density verifies a parabolic equation. We estimate $v$ in $L^\infty$ norm which enables us to control the vacuum on the density.
The Boutet de Monvel calculus of pseudo-differential boundary operators is generalised to the full scales of Besov and Triebel--Lizorkin spaces (though with finite integral exponents for the latter). The continuity and Fredholm properties proved here extend those previously obtained by Franke and Grubb, and the results on range complements of surjectively elliptic Green operators improve the earlier known, even for the classical spaces with $1<p<\infty$. The symbol classes treated are the uniformly estimated ones. Some precisions are given for the general definitions of trace and singular Green operators of class 0.
We report synthesis and magnetization of Sr2RuREO6 (RE/Ru-211O6) with RE = Gd and Eu. Both Gd,Eu/Ru-211O6 are formed in distorted perovskite-type single phase with monoclinic P21/n space group. Magnetization (M/T) measurements exhibited antiferromagnetic ordering of Ru moments for both Gd and Eu compounds at TN of around 30 K due to super-exchange Ru-O-O-Ru interaction. Interestingly enough, detailed M/H plots of Gd/Ru-211O6 exhibited, the development of a ferromagnetic (FM) component at 20 K. Further at below 10 K, the FM component disappears and rather an spin freezing field (HSF) of around 1000 Oe is seen at 5 K. In case of Eu/Ru-211O6, the basic antiferromagnetic ordering of Ru moments remains invariant down to 5 K, and the appearance of FM component is not seen. It seems that the high magnetic moment of 8mB for Gd3+ influences the AFM ordering of Ru in case of Gd/Ru-211O6.
We use a multivariate version of Stein's method to establish a quantitative Lindeberg CLT for the Fourier transforms of random $N$-vectors. We achieve this by deducing a specific integral representation for the Hessian matrix of a solution to the Stein equation with test function $e_t(x) = \exp(- i \sum_{k=1}^N t_k x_k)$, where $t,x \in \mathbb{R}^N$.
We consider a mixture of a single-component Bose gas and a two-component Fermi gas at temperatures where the Bose gas is almost fully condensed. In such a mixture, two fermionic atoms can interact with each other by exchanging a phonon that propagates through the Bose condensate. We calculate the interaction potential due to this mechanism and determine the effective s-wave scattering length for two fermions that interact, both directly by the interatomic potentials, as well as by the above mentioned exchange mechanism. We find that the effective scattering length is quite sensitive to changes in the condensate density and becomes strongly energy dependent. In addition, we consider the mechanical stability of these mixtures and also calculate the dispersion and the damping of the various collisionless collective modes of the gas.
This paper discusses the unique continuation principal of the solutions of the following perturbed fourth order elliptic differential operator $\mathcal{L}_{A,q}u=0$, where \[ \mathcal{L}_{A,q}(x,D)\ =\ \sum_{j=1}^nD^4_{x_j} + \sum_{j=1}^n A_jD_{x_j} + q, \qquad (A, q) \in W^{1,\infty}(\Omega,\mathbb{C}^n) \times L^{\infty}(\Omega,\mathbb{C}) \] whose principal term is not given by some integer power of the Laplacian operator. We derive some suitable Carleman estimates which is the main tool to prove the unique continuation principle. As a by-product, we also deduce some stability estimate and prove the strong unique continuation principle in $2$-dimension.
We present the CP-violating Next-to-2-Higgs-Doublet Model (CN2HDM) which is based on the extension of the CP-violating 2-Higgs-Doublet-Model (C2HDM) by a complex singlet field that obeys a discrete $\mathbb{Z}_2$ symmetry. The model thus features not only CP violation required for successful electroweak baryogenesis but also a Dark Matter (DM) candidate. The model has an extended Higgs sector with four CP-mixing visible neutral Higgs bosons, a DM candidate and a pair of oppositely charged Higgs bosons. The possibility of singlet and CP-odd admixtures to the observed Higgs boson in addition to the large number of visible scalar particles leads to an interesting Higgs phenomenology. We find that the model can easily provide 100\% of the DM relic density and investigate interesting LHC and DM observables within the model. We provide all the tools necessary to study the CN2HDM in detail and point out future research directions for this interesting benchmark model that can address some of the most pressing open questions of the Standard Model.
The inhomogeneous single-, two- and three-boson realizations of the more general polynomial angular momentum algebra SU_n(2) are obtained from the Fock representations of SU_n(2) that corresponds to the indecomposable representation on the space of universal enveloping algebra U(SU_n(2)) and to the induced representations on the quotient spaces U(SU_n(2))/I_i, with I_i being some left ideals of U(SU_n(2)).