Simple Gaussian Form Research Articles

Consistency relations for large-scale structures with primordial non-Gaussianities

We investigate how the consistency relations of large-scale structures are modified when the initial density field is not Gaussian. We consider both scenarios where the primordial density field can be written as a nonlinear functional of a Gaussian field and more general scenarios where the probability distribution of the primordial density field can be expanded around the Gaussian distribution, up to all orders over ${\ensuremath{\delta}}_{L0}$. Working at linear order over the non-Gaussianity parameters ${f}_{\mathrm{NL}}^{(n)}$ or ${S}_{n}$, we find that the consistency relations for the matter density fields are modified as they include additional contributions that involve all-order mixed linear-nonlinear correlations $⟨\ensuremath{\prod}{\ensuremath{\delta}}_{L}\ensuremath{\prod}\ensuremath{\delta}⟩$. We derive the conditions needed to recover the simple Gaussian form of the consistency relations. This corresponds to scenarios that become Gaussian in the squeezed limit. Our results also apply to biased tracers and velocity or momentum cross-correlations.

Theoretical analysis of the astrophysical S-factor for the capture reaction α + d → 6Li + γ in the two-body model

Theoretical estimates for the astrophysical S-factor and the rate of the reaction d(α, γ)6Li were obtained on the basis of the two-body model involving an α−d potential that has a simple Gaussian form and which describes correctly S-, P-, and D-wave phase shifts, the binding energy, and the asymptotic normalization coefficient for the S-wave bound state. The wave functions for the bound and continuum channels were calculated with the aid of the highly precise Numerov algorithm. The results for the contributions of the E1 and E2 transition components reveal a good convergence as the upper limit in the effective integrals increases up to 40 fm. The results obtained for the astrophysical S-factor and the rate of the reaction d(α, γ)6Li in the temperature range of 106 K ≤ T ≤ 1010 K agree well with the results of the calculations performed by A.M. Mukhamedzhanov and his coauthors [Phys. Rev. C 83, 055805 (2011)] by using the known asymptotic form of the wave function at low energies and a complicated two-body potential at higher energies.

The density in the density of states method

It has been suggested that for QCD at finite baryon density the distribution of the phase angle, i.e. the angle defined as the imaginary part of the logarithm of the fermion determinant, has a simple Gaussian form. This distribution provides the density in the density of states approach to the sign problem. We calculate this phase angle distribution using i) the hadron resonance gas model; and ii) a combined strong coupling and hopping parameter expansion in lattice gauge theory. While the former model leads only to a Gaussian distribution, in the latter expansion we discover terms which cause the phase angle distribution to deviate, by relative amounts proportional to powers of the inverse lattice volume, from a simple Gaussian form. We show that despite the tiny inverse-volume deviation of the phase angle distribution from a simple Gaussian form, such non-Gaussian terms can have a substantial impact on observables computed in the density of states/reweighting approach to the sign problem.

Triplet correlations and bridge functions in classical density functional theory for liquid water

Density functional theory for molecular fluids developed by Donley et al. (J. Chem. Phys. 101 (1994) 3205) is extended to include the effects of orientation-dependent bridge functions associated with the inter-particle, triplet correlations. Resultant integral equations for the pair and direct correlation functions are solved for water, where the three-body direct correlation functions are approximated in terms of two-body functions. A test calculation employing a simple Gaussian form for the two-body function between the oxygen sites then provides a promising result to improve the description of the oxygen–oxygen correlations in liquid water at room temperature.

Meta-analysis of diagnostic studies based upon SROC-curves: a mixed model approach using the Lehmann family

Meta-analysis of diagnostic studies experiences the common problem that different studies might not be comparable since they have been using a different cut-off value for the continuous or ordered categorical diagnostic test value defining different regions for which the diagnostic test is defined to be positive. Hence specificities and sensitivities arising from different studies might vary just because the underlying cut-off value had been different. To cope with the cut-off value problem, interest is usually directed towards the receiver operating characteristic (ROC) curve which consists of pairs of sensitivities and false positive rate (1–specificity). In the context of meta-analysis, one pair represents one study and the associated diagram is called SROC curve where the S stands for ‘summary’. The paper will consider—as a novel approach—modelling SROC curves with the Lehmann family that assumes log-sensitivity is proportional to the log-false positive rate across studies. The approach allows for study-specific false positive rates which are treated as (infinitely many) nuisance parameters and eliminated by means of the profile likelihood. The adjusted profile likelihood turns out to have a simple univariate Gaussian structure which is ultimately used for building inference for the parameter of the Lehmann family. The Lehmann model is further extended by allowing the constant of proportionality to vary across studies to cope with unobserved heterogeneity. The simple Gaussian form of the adjusted profile likelihood allows this extension easily as a form of a mixed model in which unobserved heterogeneity is incorporated by means of a normal random effect. Some meta-analytic applications on diagnostic studies including brain natriuretic peptides for heart failure, alcohol use disorder identification test (AUDIT) and the consumption part of AUDIT for detection of unhealthy alcohol use as well as the mini-mental state examination for cognitive disorders are discussed to illustrate the methodology.

Dynamical effects of QCD in q2 $ \bar{{q}}^{{2}}_{}$ systems

We study the coupling of a tetraquark system to an exchanged meson-meson channel, using a pure gluonic theory based four-quark potential {\em matrix} model which is known to fit well a large number of data points for lattice simulations of different geometries of a four-quark system. We find that if this minimal-area-based potential matrix replaces the earlier used simple Gaussian form for the gluon field overlap factor $f$ in its off-diagonal terms, the resulting $T$-matrix and phase shifts develop an angle dependence whose partial wave analysis reveals $D$ wave and higher angular momentum components in it. In addition to the obvious implications of this result for the meson-meson scattering, this new feature indicates the possibility of orbital excitations influencing properties of meson-meson molecules through a polarization potential. We have used a formalism of the resonating group method, treated kinetic energy and overlap matrices on model of the potential matrix, but decoupled the resulting complicated integral equations through the Born approximation. In this exploratory study we have used a quadratic confinement and not included the spin-dependence; we also used the approximation of equal constituent quark masses.

Propagation of radiation in time-dependent three-dimensional random media

In Ref. [1] (Appendix A) we derived equations governing the frequency and spatial spectrum of radiation propagating in three-dimensional time-dependent random media with randomly varying sound speed c ( x , t). From the spectral equations we determine equations for the energy flux in both the forward and backward directions. We consider media that are spatially homogeneous and isotropic and stationary in time. In order to allow an independence assumption the analysis is restricted to fluctuations that satisfy the conditions τμ ≫ L z /c 0 and τμ ≪ L FS/c 0 where τμ is the characteristic time of the fluctuations, k 0 is the mean radiation wavenumber, L z is the characteristic correlation length of the random fluctuations in the mean propagation direction and L FS is a mean scattering length. We consider various values of γ = (k 0 L z )2/2. When γ ≪ 1 we find the usual radiation transfer equations. When γ ≫ 1, but back-scattering can be neglected, we find the forward-scattering equations. We also consider γ ≫ 1, when back-scattering cannot be neglected. We consider as initial boundary conditions a plane wave and an infinite incoherent source. We present numerical solutions for γ ≪ 1, γ = O (1) and γ ≫ 1 using a simple Gaussian form for the fluctuation correlation function.

Injection spectrum of ultrahigh energy cosmic rays in the top-down scenario

We analyze the uncertainties involved in obtaining the injection spectra of UHECR particles in the top-down scenario of their origin. We show that the DGLAP $Q^2$ evolution of fragmentation functions (FF) to $Q=M_X$ (mass of the X particle) from their initial values at low $Q$ is subject to considerable uncertainties. We therefore argue that, for $x\lsim 0.1$ (the $x$ region of interest for most large $M_X$ values of interest, $x\equiv 2E/M_X$ being the scaled energy variable), the FF obtained from DGLAP evolution is no more reliable than that provided, for example, by a simple Gaussian form (in the variable $\ln(1/x)$) obtained under the Modified Leading Log Approximation (MLLA). Additionally, we find that for $x\gsim0.1$, the evolution in $Q^2$ of the singlet FF, which determines the injection spectrum, is ``minimal'' -- the singlet FF changes by barely a factor of 2 after evolving it over $\sim$ 14 orders of magnitude in $Q\sim M_X$. We, therefore, argue that as long as the measurement of the UHECR spectrum above $\sim10^{20}\ev$ is going to remain uncertain by a factor of 2 or larger, it is good enough for most practical purposes to directly use any one of the available initial parametrisations of the FFs in the $x$ region $x\gsim0.1$ based on low energy data even without evolving them to the requisite $Q^2$ value.

Diffusion quantum Monte Carlo study of three-dimensional Wigner crystals

We report diffusion quantum Monte Carlo calculations of three-dimensional Wigner crystals in the density range ${r}_{s}=100--150.$ We have tested different types of orbital for use in the approximate wave functions but none improve upon the simple Gaussian form. The Gaussian exponents are optimized by directly minimizing the diffusion quantum Monte Carlo energy. We have carefully investigated and sought to minimize the potential biases in our Monte Carlo results. We conclude that the uniform electron gas undergoes a transition from a ferromagnetic fluid to a body-centered-cubic Wigner crystal at ${r}_{s}=106\ifmmode\pm\else\textpm\fi{}1.$ The diffusion quantum Monte Carlo results are compared with those from Hartree-Fock and Hartree theory in order to understand the role played by exchange and correlation in Wigner crystals. We also study ``floating'' Wigner crystals and give results for their pair-correlation functions.

Realistic ghost state: Pauli forbidden state from rigorous solution of theαparticle

The antisymmetrization of the composite particles in cluster model calculations manifests itself in Pauli forbidden states (ghost states), if one restricts oneself on undeformed cluster in the low energy region. The resonating group method and the generating coordinate method rely on a property of the norm kernel, which introduces some of the ghost states. The norm kernel has been usually calculated under the assumption that the inner wavefunctions have a simple Gaussian form. It is the first time that this assumption is tested by the rigorous way. In the 4He+N system, we demonstrate a ghost state, which is calculated from a rigorous solution of Yakubovsky equations for the alpha particle. The ghost states calculated by rigorous and approximate methods turn out to have a very similar form. It is analytically proved that the trace of the norm kernel does not depend on the inner wavefunction we choose.

PRINCIPAL CURVES: A NEW TECHNIQUE FOR INDIRECT AND DIRECT GRADIENT ANALYSIS

Principal curves are smooth one-dimensional curves in a high dimensional space that are ideally suited for indirect gradient analysis of multispecies abundance data. A principal curves ordination will simultaneously estimate the species response curves and locate sites on a single ecological gradient. By means of theoretical argument and simulations, they are shown to be superior to both correspondence analysis and multidimensional scaling, outperforming them in 77% and 72% of simulations, respectively. The species response curves used in the simulations varied from simple Gaussian form with equal maxima and tolerances to complex multimodal curves with varying maxima and tolerances. Simulations were conducted both with and without noise. When species response curves are smooth, and a reasonable initial configuration is provided, principal curve gradient analysis can succeed even when the curves are complex and beta diversity is high. Principal curves can also be adapted for direct gradient analysis in order to relate species composition to environmental variables. Although ordination techniques are used both to uncover ecological gradients and to represent species composition, it is argued that these two aims are distinct. Hence, a single ordination technique cannot generally achieve both objectives simultaneously. However, by superimposing a principal curve on a principal components biplot, joint representation of an ecological gradient and species composition can be achieved. Information from either an indirect or direct principal curve gradient analysis can be added to the biplot, thereby relating environmental variables to species composition and locations of sites on the gradient. Two ecological data sets, comprising abundances of hunting spiders and species of grasses, are analyzed using principal curve gradient analysis. The results are contrasted with previous analyses, using canonical-correspondence analysis and canonical-correlation analysis, and indicate that principal curve gradient analysis can find one-dimensional gradients that explain species composition as well as, or better than, higher dimensional solutions from other techniques. This, in turn, can lead to a more succinct representation and better understanding of ecological systems.

Surface evaluation of ultrathin platinum films

Ultrathin platinum films deposited by electron cyclotron resonance (ECR) sputtering on silicon substrates have curious X-ray reflection properties. They appear to be rough and thick for X-rays incident below the critical angle of platinum, but smooth and thin above the critical angle. Vacuum-evaporated (VE) ultrathin platinum films do not exhibit such a property. If it is assumed that the frequency distribution D( z) of surface heights has a gaussian form when the surface profile is defined by z = f( x, y), the calculated X-ray reflectivities of VE films are very close to experimental figures, but quite different for ECR-sputtered ultrathin films. The actual surface height distributions of ECR-sputtered ultrathin films are derived from scanning tunnelling micrograph images. These distributions are produced by the growth of isolated hills, with result from ion bombardment in the ECR plasma. Because the hill makes a large shaded area on the surface, which does not contribute to X-ray reflection below the critical angle, the surface height distribution apparently changes. The transformed distribution does not have a simple gaussian form, and therefore the X-ray reflectivity of ECR-sputtered ultrathin films cannot be explained by a simple gaussian approximation.

Note on the Nuclear α-Particle Model Wavefunction with a Simple Gaussian Form

The wavefunctions with a simple Gaussian form for the α-particle in the light nuclei have been used in some literatures. This note shows that this type of the α-particle wavefunction can not fit the experimental data for electron scattering.

Effect of elastic bending energy on the emulsification failure in a microemulsion

Small-angle neutron scattering experiments have been carried out to study the phase stability of a very dilute water-in-oil microemulsion. At the fixed water to surfactant molar ratio of ω = 30, we observed a decrease in the microemulsion droplet radius and an increase in the polydispersity when the dispersed-phase volume fraction, φ, was dilute to less than 1%. Upon further dilution to below 0.1%, the droplets disappeared. This suggested that in the concentration range, 0.1% ⩽ φ ⩽ 1.0%, a single-phase microemulsion was unstable, and a phase separation, known as the emulsification failure, occurred. To understand the interesting phase behavior, we have formulated an analytic solution for the droplet size distribution function for a two-phase microemulsion based on the theory of Borkvec et al. ( J. Colloid Interface Sci. 131, 366 (1989)). We found that the distribution function attained a simple Gaussian form, and the width of the distribution function was inversely proportional to the renormalized elastic bending energy of the microemulsion droplets. Other features predicted by the theory were also in good agreement with the small-angle neutron scattering measurements.

Nucleon-nucleon scattering in the 0-6 GeV range and the relativistic optical model based on deep attractive forbidden state potentials.

The optical relativistic model based on deep attractive forbidden state quasipotentials describes well the angular and energy dependence of differential np- and pp-scattering cross sections and polarizations, including the transition from the U-shaped form of the angular distributions of np scattering to forward scattering at the energy ${\mathit{E}}_{\mathrm{lab}}$\ensuremath{\ge}1 GeV, when the scattering P-phase shifts, equal to \ensuremath{\pi} at low energies, pass through \ensuremath{\pi}/2 (the triplet and singlet S-phase shifts start from the values 2\ensuremath{\pi} and \ensuremath{\pi}, respectively). The higher scattering phase shifts (L\ensuremath{\ge}2) are small everywhere. The potentials are determined from the scattering phase shifts in the low-energy region ${\mathit{E}}_{\mathrm{lab}}$1 GeV. They have been chosen in the simple Gaussian form and for different partial waves they differ in depth and width (from ${\mathit{V}}_{0}$=0.73 GeV, a=0.85 fm to ${\mathit{V}}_{0}$=2.40 GeV, a=0.45 fm). A few negative phase shifts $^{3}$${\mathit{D}}_{1}$, $^{3}$${\mathit{F}}_{3}$, $^{3}$${\mathit{G}}_{3}$, which reflect the peripheral repulsion due to the spin-orbital and tensor interactions, are calculated by means of the one-boson-exchange potential periphery matched to the central attraction. The imaginary part W of the optical potential V+iW is determined by the value ${\mathrm{\ensuremath{\sigma}}}_{\mathrm{tot}}$/${\mathrm{\ensuremath{\sigma}}}_{\mathrm{el}}$ and grows rapidly with increase in the energy ${\mathit{E}}_{\mathrm{lab}}$, so that the phase-shift values lose sensitivity to the real part V of optical potential if ${\mathit{E}}_{\mathrm{lab}}$ exceeds 5 GeV. Finally, the quantum chromodynamics (QCD) effects are discussed which may underlie the potentials considered.

Modeling collisional energy transfer in highly excited molecules

Data from classical trajectory simulations of the collision of a highly excited molecule with a monatomic bath gas are used to test the validity of the precepts used in the biased-random-walk (BRW) model for collisional energy transfer. This model assumes that energy migration during the collision is pseudorandom except for the constraint of microscopic reversibility, and leads to a simple displaced Gaussian form for the energy-transfer probability distribution. The BRW assumptions are shown to be of acceptable validity to exact classical trajectory simulations. A simple analytical approximation to the mean-square energy transfer per collision is obtained which reproduces the trajectory data to within an average of ±20%, and also gives acceptable accord with experimental data. The model shows that the magnitude of the average energy transferred per collision is governed by the time taken to traverse the overall interaction potential in and out from the appropriate collision diameter, by the internal energy, and by the average force exerted at the classical turning point of individual reactant-atom–bath-gas interactions.

Calibration of Si(Li) detectors for X-ray astronomy

The characterization of a high performance, energy dispersive Si(Li) solid-state detector is presented. The response of the detector to K and L lines in the energy range 0.7–9.0 keV is obtained by using a Van de Graaff accelerator to bombard selected targets having 12 ⩽ Z ⩽ 50 with a high energy proton beam. In addition to the channel-to-energy calibration and resolution vs energy plot, an empirical relation is presented for line yield as a function of energy for the K lines. Further, spectral data are used to estimate the silicon dead layer thickness. Finally, a detailed characterization of the detector's response function, including deviations from a simple Gaussian form due to effects such as incomplete charge collection, is given. From these results, a procedure is indicated by which X-ray spectrometers used in X-ray astrophysics applications may be calibrated.

The large-scale structure of homogenous turbulence

The large-scale structure of homogenous turbulence