Longitudinal and transverse structure functions in decaying nearly homogeneous and isotropic turbulence

Streamwise evolution of longitudinal and transverse velocity structure functions in a decaying homogeneous and nearly isotropic turbulence is reported for Reynolds numbers Reλ up to 720. First, two theoretical relations between longitudinal and transverse structure functions are examined in the light of recently derived relations and the results show that the low-order transverse structure functions can be well approximated by longitudinal ones within the sub-inertial range. Reconstruction of fourth-order transverse structure functions with a recently proposed relation by Grauer et al. is comparatively less valid than the relation already proposed by Antonia et al. Secondly, extended self-similarity methods are used to measure the scaling exponents up to order eight and the streamwise evolution of scaling exponents is explored. The scaling exponents of longitudinal structure functions are, at first location, close to Zybin's model, and at the fourth location, close to She–Leveque model. No obvious trend is found for the streamwise evolution of longitudinal scaling exponents, whereas, on the contrary, transverse scaling exponents become slightly smaller with the development of a steamwise direction. Finally, the stremwise variation of the order-dependent isotropy ratio indicates the turbulence at the last location is closer to isotropic than the other three locations.

Analysis of the Longitudinal Structure Function F_L from the Non-linear Regge Gluon Density Behavior at Low-x

The longitudinal structure function with shadowing correction according to the nonlinear effects of the gluon density behavior at low x is considered. The solution of the GLR-MQ evolution equation for the gluon density shows that the FL^g（x, Q2） behavior can be tamed by the singularity at low x values. Comparing our results with H1 data at R=4 GeV-1 shows that at very low x this behavior is completely tamed by taking shadowing correction into account.

A Lower Bound on Longitudinal Structure Function at Small x from a Self-Similarity Based Model of Proton

Self-similarity based model of proton structure function at small x was reported in the literature sometime back. The phenomenological validity of the model is in the kinematical region 6.2 × 10-7 ≤ x ≤ 10-2 and 0.045 ≤ Q2 ≤ 120 GeV2. We use momentum sum rule to pin down the corresponding self-similarity-based gluon distribution function valid in the same kinematical region. The model is then used to compute bound on the longitudinal structure function FL（X, Q2） for A1tarelli-Martinelli equation in QCD and is compared with the recent HERA data.

Importance of heavy quark longitudinal structure function measurements at future circular collider energies

In this article,we consider the ratio of structure functions for heavy quark pair production at low values of.The importance of this ratio for charm and beauty pair production is examined according to the Hadron Electron Ring Accelerator(HERA)data.The behavior of these ratios is considered due to the hard pomeron behavior of the gluon distribution function.The results are in good agreement with the HERA data.Expanding this data to the range of new energies underscores the importance of these measurements for heavy quarks.The ratio of charm and beauty structure functions at the proposed Large Hadron electron Collider(LHeC)is considered as a function of invariant center-of-mass energy.For top pair production this ratio is extracted with known kinematics of the LHeC and Future Circular Collider electron-hadron(FCC-eh)colliders.Comparison of the results obtained for the ratio of top structure functions in LHeC and FCC-eh are proportional to the specified inelasticity range.

Generalised variance functions for longitudinal survey data

In this research, we propose longitudinal generalised variance functions (LGVFs) to produceconvenient estimates of variances by incorporating time effect into modelling. Asymptoticproperties of some certain type of estimators are investigated. Simulation studies and implementation of the proposed methods to Current Population Survey (CPS) data show that LGVFswork well in producing standard error estimates.

Weighted quantile regression for longitudinal data using empirical likelihood

This paper proposes a new weighted quantile regression model for longitudinal data with weights chosen by empirical likelihood(EL). This approach efficiently incorporates the information from the conditional quantile restrictions to account for within-subject correlations. The resulted estimate is computationally simple and has good performance under modest or high within-subject correlation. The efficiency gain is quantified theoretically and illustrated via simulation and a real data application.